Washington (CNN) President Donald Trump's attorney Rudy Giuliani said Wednesday that he wants negotiations over a presidential interview with special counsel Robert Mueller to be completed by Sept. 1. But the presidential attorney is also making the case that Republicans could benefit from dragging the process into the November midterm elections.

"When I first got involved, I would have told you not testifying would be the right legal strategy but then hurt politically," Giuliani told CNN in a phone interview Wednesday. "Now I'm thinking the continuance of the investigation would actually help because people are getting tired of it, and (the President) needs something to energize his voters because the Democrats look like they're energized. Nothing would energize (Republicans) more than, 'Let's save the President.' "

Giuliani's comments came as the Trump legal team sent its latest rebuttal to the special counsel's office, once again laying out concerns with the scope of Mueller's request to interview the President in a letter that appears to do little to bridge the gap with Mueller's team.

Republican operatives increasingly view the special counsel's investigation as a base-rallying tool for the midterms -- an outlook reflected in Giuliani's musings about interview negotiations bleeding into the fall campaign.

But his comments are also a reminder of how little the President's legal team knows about how Mueller intends to proceed.

Mueller wants to question Trump about his campaign's interactions with Russia during the campaign and allegations of obstruction of justice leveled against the President, sources familiar with the matter say. The Trump team has been steadfast in arguing the latter topic should be out of bounds because it pertains to privileged matters during his time as President.

The reality -- which Giuliani readily conceded -- is that Mueller is in control of the timing here, not them.

"Mueller will do whatever he wants to do. He has a game plan. He knows it. We just don't know it," Giuliani said.

Trump's team braces for a long battle

Trump's legal team has argued for months that Mueller has all the information he needs to conclude the investigation by relying on the documents provided by the White House, contemporaneous notes and interviews with people inside and outside the White House. They believe that unless Mueller can establish some compelling reason for Trump to testify at all, there is at least some hope that Deputy Attorney General Rod Rosenstein would decline to approve a subpoena of the President.

In the meantime, all Trump's team can do is continue negotiations, as it did Wednesday with yet another counteroffer to Mueller intended to once again narrow the scope of the questions he would ask the President in any interview, according to sources familiar with the matter.

"Millions of pages of documents along with testimony from dozens of witnesses have been provided. We're re-stating what we have been saying for months: it is time for the Office of Special Counsel to conclude its inquiry without further delay," Giuliani said of the letter in a statement on Wednesday.

A source familiar with the discussions also reiterated that the Trump team is still "not inclined to do an oral interview." Despite this, the President remains willing to talk to the special counsel, his lawyers say.

The source declined to delve into the details of the letter but said the legal team expressed its "concerns" about Mueller's parameters for an interview.

As they await a response from Mueller, Trump's legal team appears prepared for a prolonged battle even as Giuliani publicly declares it could all wrap up in early September.

A source familiar with the legal discussions noted that negotiations over prosecutors' questioning of a President have historically been a lengthy process, pointing to independent counsel Ken Starr's efforts to get an interview with President Bill Clinton.

The Clinton negotiations lasted six months and involved more than a dozen letters between the two sides. By comparison, negotiations with Mueller's team over an interview restarted in earnest with the most recent iteration of Trump's legal team more than three months ago.

Gaming out how Mueller -- and Trump -- will respond

How long the tug of war over a presidential interview lasts could rest on Mueller and Rosenstein's willingness to subpoena the President if Trump's legal team refuses to budge.

Mueller has never explicitly given the Trump legal team a timeline for reaching that point. While he has floated the possibility of subpoenaing Trump, he has never explicitly threatened to do so, and the Trump team believes he will not pursue that option. Working through the federal appeals courts to reach a final decision on the merits of a presidential subpoena at the Supreme Court could take months.

Again, Trump's lawyers admit, they're just guessing -- rolling the dice that Mueller would rather avoid that legal fight if he can, but no one can say for sure that he won't do it, sources with knowledge of their thinking said.

For now, Mueller appears to be taking his time. When the President's lawyers responded to the special counsel the last time, their response sat unanswered for weeks, sources familiar with the negotiations tell CNN.

When Mueller finally responded, the offer narrowed the number of questions but didn't move forward much on limiting the scope of the questioning, according to sources familiar with the President's attorneys' thinking.

Wednesday's new counteroffer is intended to find more common ground, but it is still unclear -- and some think unlikely -- that they will find agreement on questions pertaining to obstruction of justice that Mueller wants to ask the President.

Trump's lawyers consistently argue that they will not subject the President to what they call a "perjury trap."

"For example, 'What did you say about Flynn?' 'Why did you fire Comey?' They already know our answer," Giuliani said.

There is an additional unpredictable factor here: the President himself. He has publicly said he wants to testify, and those with knowledge of his views insist he still does. How strongly he feels about that is unknown.

"Ultimately, this is the President's decision," Jay Sekulow, one of Trump's attorneys, said on his radio show Wednesday alongside Giuliani. "But we're hopeful that he will take the advice of his lawyers as this process continues to mature."

But after all these months, the timing of this -- and whether Mueller would subpoena a sitting President -- remains a guessing game.

Giuliani believes it would be a mistake for Mueller to carry this through Sept. 1 and as the election approaches.

"He will become an election issue, like it did for Comey," Giuliani said. "I don't think he thinks that way, but I think he cares about his reputation."