(CNN) The White House is fighting to regain control of the confirmation of Supreme Court pick Brett Kavanaugh, but a controversy is widening over its role in an FBI investigation into allegations against the nominee of sexual assault and misconduct.

New signs emerged Sunday that the administration is taking steps to make sure the probe is narrow and focused -- as Republicans seek to forestall further delays to the confirmation process or another erosion of the judge's position.

So while President Donald Trump may have been forced to reopen an FBI background check on Kavanaugh, it is clear he is determined to get the process back on track in a strategy already inflaming partisan discord over the nomination.

A swift investigation by the FBI -- should it produce no new information about Kavanaugh's past that hurts his chances -- could permit Republicans to fire up the machinery of the Senate and potentially call procedural votes to advance the nomination by the end of the week.

Such a timetable would keep alive the effort to present GOP base voters with a generational deliverable and a solid conservative majority on the Supreme Court that could juice turnout for midterm elections in November that are looking favorable for Democrats.

But the extent to which the White House is controlling the process is likely to stoke fresh turmoil around a nomination that is already certain to trigger long-term political reverberations.

The investigation is the result of a dramatic day in Washington on Friday that began with quick movement toward Kavanaugh's confirmation and ended with a pause in the process while the FBI steps in. Following the agreement for the FBI to investigate Kavanaugh, the focus has moved to how the probe will be handled, its ultimate findings, and whether it will put to rest fears about a lack of due diligence over the nominee for the nation's highest court.

Trump said Saturday that the FBI would have "free rein" to investigate whatever it wanted in the probe, saying, "They have free rein to do whatever they have to do." But potential inconsistencies emerged Sunday between public statements by administration officials that appeared to finesse Trump's assertion and what some sources are saying privately about the probe.

Trump's counselor Kellyanne Conway told CNN's Jake Tapper on "State of the Union" that the reopened FBI checks into Kavanaugh would be "limited in scope, it's meant to last one week, and ... it's not meant to be a fishing expedition ."

But she also insisted that Trump respects the independence of the FBI and believes it should look into "anything that is credible within that limited scope."

White House spokeswoman Sarah Sanders said on "Fox News Sunday," meanwhile, that "the White House is not micromanaging this process."

But she also warned that "Senate Republicans are going to lay out and dictate those terms, and we look forward to this wrapping up so we can see what was seen in the last six investigations that Judge Kavanaugh has been a part of."

A source with knowledge of the investigation told CNN that the FBI will interview a handful of people and only take direction from the White House, not the Senate.

A source briefed on the matter said the White House sent the limited parameters of the investigation to the FBI based on a Senate request.

Two sources with knowledge of the situation told CNN, however, that Senate Republicans were working with White House counsel Don McGahn, who was "trying to make it as narrow as possible."

Democrats concerned

Indications that the White House was intimately involved in the process could risk tainting its eventual conclusions or leave the impression that the probe is using the prestige of the FBI but is only a cursory effort designed to restore Kavanaugh's momentum as quickly as possible.

Trump was quick to try to scotch suggestions that the White House was taking an inappropriate interest in the investigation -- doing so in a way that hinted at how the storyline could explode in the days to come.

"Wow! Just starting to hear the Democrats, who are only thinking Obstruct and Delay, are starting to put out the word that the 'time' and 'scope' of FBI looking into Judge Kavanaugh and witnesses is not enough. Hello! For them, it will never be enough - stay tuned and watch!" the President tweeted.

The investigation only happened at all because Arizona Sen. Jeff Flake blinked last week and made his vote to confirm Kavanaugh contingent on further inquiries into the allegations.

Signs that the White House is more closely involved in setting the terms of the new background check on Kavanaugh than might have been initially clear sparked disquiet from Democrats.

"Based on some of the reports we've seen this weekend, I'm very concerned about this because the White House should not be allowed to micromanage an FBI investigation," Democratic Sen. Amy Klobuchar of Minnesota, who sits on the Senate Judiciary Committee, said on CNN's "State of the Union."

"While the White House decides who to nominate, and then that person is submitted to a background check, I have never heard that the White House, either under this president or other presidents, is saying, 'well, you can't interview this person, you can't look at this time period, you can only look at these people from one side of the street from when they were growing up,'" Klobuchar said.

Asked on ABC's "This Week" whether she thought the FBI investigation would be credible, Hawaii Democratic Sen. Mazie Hirono said, "That's going to be the big outstanding question."

"I would think that Jeff Flake and the other senators who are going for this investigation will call for -- there's time, you know," said Hirono, who is also a Judiciary Committee member. "The thing is that every Senate vote matters and there is ... time to get to the bottom of it, even if it's seven days.

"That's bad enough," Hirono continued. "But then to limit the FBI as to the scope and who they're going to question, that, that really -- I wanted to use the word farce -- but that's not the kind of investigation that all of us are expecting the FBI to conduct."

FBI's witness list

The latest White House gambit will also, once again, drag the FBI into the heat of the political battle in Washington -- an unfortunate pattern that has repeated itself over the last few years including with the Clinton email controversy, the Russia investigation and Trump's constant pressure on the bureau.

Initially, it appears, the bureau will talk to people who could shed light on Christine Blasey Ford's allegation that a drunken Kavanaugh sexually assaulted her at a house party when they were teenagers. This is expected to include Mark Judge, whom Ford alleges was in the room at the time, and two other people Ford said were at the party, Patrick J. Smyth and Leland Keyser, sources familiar with the matter tell CNN. Judge has already said that he had no memory of any alleged incident involving Ford and Kavanaugh, and Smyth and Keyser have said they don't remember the party. However, Keyser, a friend of Ford's, does not refute the allegation and has said she believes the account , her lawyer said in a letter to the Senate Judiciary Committee.

But Ford and Kavanaugh -- who gave testimony under oath to the committee on Thursday -- were not on the initial list of names given by Senate Republicans to the White House, according to two sources familiar with the matter.

Questioning Judge could go some way to easing concerns among several Republicans about the questions surrounding the Ford allegations. Unless he changes his story, however, it seems unlikely that the FBI investigations produce new actionable evidence that could potentially disqualify Kavanaugh, especially since the alleged incident took place 36 years ago.

That could play into an argument Republicans have used to bolster Kavanagh's confirmation hopes: that he should not be blocked from the Court based on allegations that cannot be corroborated and were not prosecuted at the time.

Indeed, Kavanaugh's supporters could find reason for optimism in an independent assessment of Ford's allegation that the outside counsel at Thursday's GOP-led Judiciary Committee hearing sent to the Republican members of the committee Sunday night.

Rachel MItchell, the sex crimes prosecutor who was tasked with questioning Ford on her allegation of sexual assault, said a "reasonable prosecutor" would not bring a case against Kavanaugh based on Ford's allegation given the evidence presented to the committee.

Mitchell cited inconsistencies in Ford's statements to the committee, as well as to The Washington Post and her therapist, and noted the lack of corroboration of her account, including recalling details that could back her story.

"In the legal context, here is my bottom line: A 'he said, she said' case is incredibly difficult to prove," Mitchell wrote. "But this case is even weaker than that. Dr. Ford identified other witnesses to the event, and those witnesses either refuted her allegations or failed to corroborate them. For the reasons discussed below, I do not think that a reasonable prosecutor would bring this case based on the evidence before the Committee. Nor do I believe that this evidence is sufficient to satisfy the preponderance-of-the-evidence standard."

Additional accusers

Still, Democrats have argued, especially in the light of the changed politics of the #MeToo era, that anyone elevated to the Supreme Court should be above suspicion or moral reproach, and that the allegations should be given credence even if they are not prosecuted.

But the probe could potentially give Republican senators like Flake, Maine's Susan Collins and Alaska's Lisa Murkowski, who were put on the spot by Ford's emotional testimony on Thursday, more cover to vote for Kavanaugh.

A source familiar with the situation told CNN's Ariane De Vogue that the FBI had spoken on Sunday to Deborah Ramirez, another woman who came forward with allegations against Kavanaugh from his time at Yale University, which the judge has also denied.

But there is so far no indication that the FBI has or will talk with Julie Swetnick, whose attorney, Michael Avenatti, also represents Stormy Daniels, the adult film star who alleges she had an affair with Trump. Trump has denied that allegation.

Swetnick has also made allegations against a young Kavanaugh, saying he was present at a party in which she was raped, and that both he and Judge would consistently engage in excessive drinking. She did not identify either Kavanaugh nor Judge as her attackers.

Kavanaugh and Judge have emphatically denied Swetnick's allegations.

But the President slammed Avenatti as a "low life" and a "third rate lawyer" who makes false accusations.

So the chances that the White House, if it is controlling the scope of the FBI investigation into Kavanaugh, offers any opening to Avenatti appear slim.