Washington (CNN) President Barack Obama's hard road to fill a vacancy on the Supreme Court faces a pivotal week. Interest groups on the right and left are ramping up for a protracted fight, and the President will sit down with his Republican adversaries who have vowed to block his still unnamed nominee no matter who it is.

The White House said Monday it was naming Obama's senior adviser Brian Deese to lead its efforts in selecting a nominee for the court vacancy, and later navigating an already-complex confirmation process.

Deese, currently responsible for Obama's climate change agenda, previously shepherded a controversial bailout of the American auto industry and later helped manage Obama's budget office.

A lawyer with a degree from Yale Law, Deese will "ensure that the full capacity of the White House is trained on this effort, even while the rest of the White House continues its important work on other Presidential priorities," according to a White House official. The official said Deese would report to the chief of staff and ultimately Obama.

He'll join Obama's counsel Neil Eggleston and other top advisers in vetting potential candidates for the top job, a process that's expected to continue this week. Former Obama aides, including ex-senior adviser Stephanie Cutter and recently-departed legislative director Katie Fallon, will assist in presenting Obama's case to lawmakers, White House Press Secretary Josh Earnest said last week.

Obama spent this weekend reviewing material culled from public records about potential candidates for the job, a process he began a week earlier.

High-stakes meeting with McConnell, Grassley on Tuesday

In a critical session Tuesday, Obama will meet at the White House with Senate Majority Leader Mitch McConnell and Senate Judiciary Committee Chairman Chuck Grassley, the influential Republicans who announced last week they won't schedule confirmation hearings or even welcome a nominee into their offices for a traditional courtesy call.

Privately, Republicans complained the Oval Office meeting is really just a "photo-op" designed by the White House to pressure and embarrass GOP leaders for their tough and controversial stand against filling the vacancy now.

But in a joint statement issued shortly after accepting the President's invitation, McConnell and Grassley said, "We look forward to reiterating to him directly that the American people will be heard and the next Supreme Court justice will be determined once the elections are complete and the next President has been sworn into office."

Also attending will be Senate Minority Leader Harry Reid and the top Democrat on the Judiciary Committee, Sen. Patrick Leahy, who have railed against Republicans and are already leading an intense public relations battle against them to try to force them to change their minds and act on a nominee.

"I along with every other member of the Democratic caucus will be on the floor next week, the week after that, the week after that, as long as it takes to bring to the attention of America the failure of this Republican Senate to meet its constitutional mandate," Reid said.

Republicans dismissed Reid's complaints and pointed to statements in years past from senior Democrats -- such as Vice President Joe Biden -- who spoke out against approving Supreme Court justices in the last years of the terms of Republican presidents.

"So I would ask our friends across the aisle, while they come out on the floor or give press conferences and express mock horror," said Texas Sen. John Cornyn, a Judiciary Committee member and second-ranking GOP leader, "to tone down the rhetoric and avoid the hypocrisy that seems so apparent when they argue for different standards today than they advocated in the past."

Obama reached out last week to other Republican senators who are known to work across the aisle to gauge interest and shore up support for confirmation hearings and a vote on a nominee. He met at the White House with Sen. Lamar Alexander of Tennessee and spoke by phone with Sen. Jeff Flake of Arizona.

In his call with Flake, Obama expressed disappointment at Flake's opposition to considering any nominee.

Flake said Obama told him: "I hoped you guys would keep your powder dry."

"He understands, as he said publicly, the position we're in. And he understands the precedent as well. He had his own experience with Alito. That's why it becomes difficult for the Democrats to really say too much about it," said Flake referencing then Sen. Obama's vote to prevent a confirmation vote on now-Justice Samuel Alito, an appointee of President George W. Bush.

Liberal groups expect to coordinate with White House

Meanwhile, Obama continues to consider possible nominees.

One new name that surfaced over the weekend: Ketanji Brown Jackson , a judge on the U.S. District Court for the District of Columbia Circuit, the National Law Journal reported

Jackson has a unique personal connection to the congressional GOP leadership, related by marriage to House Speaker Paul Ryan.

Unlike during the period leading up to Obama's nominations of Sonia Sotomayor and Elena Kagan, most of the progressive groups with an interest in the next Supreme Court justice have spent their time not on suggesting a particular candidate but lending their support to an effort to ensure that a hearing eventually occurs.

As such, they are energizing supporters on a grass-roots level to send a message that the Senate must "do its job." They expect to coordinate their message and refrain, in general, from presenting lists of people they think should be considered.

That resolve faltered for some last week when Harry Reid came out to support Nevada Gov. Brian Sandoval, a Republican, who has been public of his criticism of the President's executive actions on immigration as well as the Affordable Care Act. Sandoval, however, quickly said he wasn't interested the nomination.

Senate Republicans look to change the subject

Despite the vocal protests from Democrats, it's not clear whether Republicans have too much to fear from them since Democrats have acknowledged they won't try to shut down the floor or use other hardball tactics that could force the GOP to relent.

In fact, Republicans plan to move this week to the first of three bipartisan bills in an effort to switch the attention from the Supreme Court standoff. In addition, Republicans believe by taking up the popular bills, it will provide further incentive for Democrats not to obstruct the Senate.

One bill addresses the growing problem with opioid addition, which is plaguing many states. A second is a broad energy modernization bill. The third is a measure that would provide loans and other funds to Flint, Michigan and other communities that have lead in their water supply and other problems with their water systems.

But Sen. Chuck Schumer said the Republicans' obstinate stand against the nominee will outweigh whatever other accomplishments the GOP tries to claim.

"At a time when Sen. McConnell has first said he wants to be the Senate for regular order, he is now going to walk around with obstructionist written on his forehead wherever he goes," said Schumer, a Judiciary Committee member and the No. 3 Democrat who is likely to succeed Reid as leader next year. "The best weapon we have is the public, and the public will put pressure on the senator, and the senators will put pressure on McConnell, and we will win this fight."

Another Judiciary Committee Republican, the quick-witted Lindsey Graham of South Carolina, said he would change his stance against acting on a nominee under one condition: "If Schumer could pass a lie detector that he would not do what we're doing if the shoe was on the other foot, I might change."