The No. 2 Senate Republican, John Cornyn, flatly disagreed with them, however, siding with Senate Majority Leader Mitch McConnell, and GOP leaders have seized on a Joe Biden speech from 1992 to make their case.
Sen. Susan Collins, a Maine Republican, told CNN that the Senate should take the process "one step at a time," but would also consider a nominee from Obama.
"For my part, it's clear that the President can send up a nominee -- regardless of where he is before he leaves office," Collins told CNN. "It is the duty of the Senate, under the Constitution, to give our advice and give our consent or withhold our consent. I believe we should follow the regular order and give careful consideration to any nominee that the president may send to the Senate."
Cornyn, a Texas Republican, was clear when asked Monday if there should be any hearings.
"No," he told CNN. "My view is that this is not about the potential nominee; it's about who chooses. And I believe strongly this should be a referendum on who chooses in November."
But Sen. Mark Kirk, an Illinois Republican facing a tough re-election bid, went a step further than Collins and said that not only should a nominee get a hearing, but also a vote -- something Senate Majority Leader Mitch McConnell has vowed to stop from happening.
"I fully expect and look forward to President Barack Obama advancing a nominee for the Senate to consider," Kirk wrote in an op-ed in The Chicago Sun-Times
Kirk and Collins' stance could complicate things for McConnell as he prepares for a showdown with the White House over filling the vacancy left on the Supreme Court by Antonin Scalia.
Senate Judiciary Committee Chairman Chuck Grassley delivered an emphatic argument against holding hearings, citing not his own reasoning, but instead a lengthy floor speech then-Sen. Joe Biden delivered in 1992. Biden's remarks excoriate the prospect of a president submitting a nomination in an election year.
"Once the political season is under way, action on a Supreme Court nomination must be put off until after the election campaign is over," Grassley said quoting the vice president, dubbing the remarks "The Biden Rules."
"(Biden) knows what the Senate should do. And I believe in his heart of hearts he understands why this Senate must do what he said it must do in 1992," Grassley said.
An administration official said Tuesday that Grassley hasn't yet taken up invitations from the White House to meet Obama for consultations on the Supreme Court pick.
On Monday, the vice president's office released a statement suggesting the situations are not comparable.
"While some say that my comments in June 1992 contributed to a more politicized nomination process, they didn't prevent the Senate from fulfilling its constitutional duties, because there was no vacancy at the time. During my career on the Judiciary Committee, I ensured the prompt and fair consideration of nine Supreme Court Justices and the current Senate has a constitutional duty to do the same," said Biden's statement.
The vice president added that his record chairing the Senate Judiciary Committee for eight years was "hard to beat."
"I presided over the process that resulted in Justice Kennedy, a Reagan nominee, being confirmed to the Supreme Court in a presidential election year. I allowed the nominations of Judge Bork and Justice Thomas to proceed to the floor, even though they didn't have the support of the committee," he said.
Senate Minority Leader Harry Reid delivered a fiery floor speech Monday, teeing off the Supreme Court nomination battle that is engulfing the Senate.
"We're seeing an unprecedented attempt to hold hostage an entire branch of government," Reid said.
More clarity about a path forward will come on Tuesday, when there will be a special meeting of Republicans members of the Judiciary Committee, according to a Grassley aide.
The meeting will meet in the Capitol shortly before Republicans host their Tuesday lunch, a McConnell aide said.
McConnell himself told reporters he would have more to say Tuesday.
Another vulnerable GOP senator up for re-election in 2016, New Hampshire's Kelly Ayotte, said she would defer to Grassley on the question of hearings.
Meanwhile, Reid met with Nevada Gov. Brian Sandoval Monday in his office, aides told CNN.
Sandoval, a moderate Republican and a former federal judge, was in town for a meeting of the National Governors Association. They discussed Sandoval's interest in a nomination, and sources familiar with the discussion say Sandoval is interested in the nomination, but Reid is not pushing the idea on him.
The political wrangling extends beyond just control of the high court, but also potentially of the Senate. Republicans hold a 54-46 seat advantage over Democrats, but Democrats would have to pick up at least four seats to win back control (assuming they keep the White House in November.) A fight over whether or not to hold a vote on an Obama nominee will likely play a role in those Senate campaigns.
Senior Democratic officials have told CNN the only way Obama wins a nomination battle is if Republicans in swing states break rank -- including Kirk and Sen. Pat Toomey of Pennsylvania.
Conversely, Senate Republicans have said that if it looks like the White House may stay in Democratic hands, they could rally conservatives.
The dynamic has already proved tricky for one of the more moderate members of the Senate Republican Conference, Alaska Sen. Lisa Murkowski. She initially said that Obama's nominee should receive a hearing, but later said Obama should hold off picking a nominee
But some Democrats have their own concerns.
Sen. Joe Manchin, the conservative West Virginia Democrat, told CNN that Democrats shouldn't tie up the floor in retaliation to a Republican blockade on a nominee.
"They know how I feel," he said of his party's leaders, adding that he believes there should be hearings and votes for the eventual nominee. "Let the President do his job."