(CNN) President Barack Obama said Wednesday it would be "difficult" for Senate Majority Leader Mitch McConnell to explain his decision not to consider a Supreme Court nominee without looking like he's motivated by politics.

Meanwhile, Senate Minority Leader Harry Reid suggested a Republican, Nevada Gov. Brian Sandoval, as a potential nominee. A source confirmed to CNN that the White House is vetting Sandoval, although he has not been actively involved in the process.

"Neither Governor Sandoval nor his staff have been contacted by or talked to the Obama Administration regarding any potential vetting for the vacancy on the U.S. Supreme Court," according to a statement from Mari St. Martin, his communications director.

Obama's pointed remarks and Reid's unconventional suggestion come amid a bitter standoff between Senate Republicans and the White House over naming a Supreme Court replacement for Justice Antonin Scalia, whose death this month launched an epic election-year fight over constitutional powers and precedent.

"I recognize the politics are hard for them because the easier thing to do is to give in to the most extreme voices within their party and stand pat and do nothing," Obama said. "But that's not our job. Our job is to fulfill our constitutional duties."

Claiming he felt sympathy for Republican lawmakers making "sheepish" arguments for blocking his court choice, Obama insisted he would "nominate somebody and let the American people decide as to whether that person is qualified."

In a blog post early Wednesday, Obama reiterated again the broad outlines of what he's looking for in a candidate to replace Scalia, despite hardening resistance among Senate Republicans toward considering his eventual Supreme Court nominee.

Writing on the SCOTUSBlog website , Obama repeated his desire for a candidate who could bring life experience to the bench, along with an unassailable job history.

"A sterling record. A deep respect for the judiciary's role. An understanding of the way the world really works. That's what I'm considering as I fulfill my constitutional duty to appoint a judge to our highest court," Obama wrote. "And as senators prepare to fulfill their constitutional responsibility to consider the person I appoint, I hope they'll move quickly to debate and then confirm this nominee so that the Court can continue to serve the American people at full strength."

Obama's entreaty to lawmakers came a day after Republicans on the Senate Judiciary Committee vowed in a letter to forgo hearings on the White House's selection, a move unprecedented in Supreme Court nomination history. McConnell showed little sign he would retreat from his position that Obama's successor should select Scalia's replacement on the high court.

"I don't know how many times we need to keep saying this: The Judiciary Committee has unanimously recommended to me that there be no hearing. I've said repeatedly and I'm now confident that my conference agrees that this decision ought to be made by the next president, whoever is elected," McConnell said Tuesday, adding later he was unlikely to even meet with Obama's nominee.

The chairman of the Judiciary panel, Republican Sen. Chuck Grassley, spoke with Obama about the selection process last week, but has so far declined invitations to meet with Obama in person, a White House official said late Tuesday.

An aide to McConnell said his office is working on arranging a meeting with Obama and senators from both parties about the Supreme Court vacancy, but if it is scheduled, he will use the meeting to tell the President directly he doesn't think a nominee should be forthcoming until a new president is in office.

"We are working with the White House to schedule a time for a meeting. (McConnell) welcomes the opportunity to reiterate to the President directly that the American people will be heard on this matter, and the nomination will be determined by whoever wins the presidency in the fall," said Don Stewart, a McConnell spokesman.

The meeting is not expected before next week, Stewart said.

However, earlier Wednesday, Sen. Patrick Leahy, the top Democrat on the Judiciary Committee, told CNN that he, McConnell, Reid and Grassley would meet with Obama on Thursday.

"He's asked me for some suggestions and I'll give them to him privately as I always have," Leahy said.

An aide to Grassley did not return an email requesting information about the meeting and whether the Iowa Republican would attend.

Speaking in the Oval Office Wednesday, Obama said none of the country's founding fathers believed a president should stop doing his job in his final year in office. And he argued that Republicans risked damaging the ability of any president to appoint judges if they proceed with blocking his Supreme Court pick.

"If, in fact, the Republicans in the Senate take a posture that defies the Constitution, defies logic, is not supported by tradition simply because of politics, then invariably what you're going to see is a further deterioration in the ability of any president to make any judicial appointments," Obama said.

"Appointments to the Supreme Court as well as the federal bench suddenly become a complete extension of our polarized politics," he added.

And he sought to dispel any damage from archival video showing Vice President Joe Biden, then a senator, arguing against approving a Supreme Court nominee during the 1992 election year.

"They've suggested there have been times that Democrats have said it would be wise for a president not to nominate someone," Obama said. "We know senators say stuff all the time."

Obama revealed little about his process in selecting a nominee, either during his remarks or in his online posting. He repeated the broad criteria for a candidate that he cited during his past two Supreme Court nomination opportunities.

"Needless to say, this isn't something I take lightly," Obama wrote. "It's a decision to which I devote considerable time, deep reflection, careful deliberation, and serious consultation with legal experts, members of both political parties, and people across the political spectrum."

But even the selection of a Republican nominee appeared unlikely to sway Senate Republicans, who in some cases said they would refuse to meet with a nominee.

Sen. Orrin Hatch, a Utah Republican and veteran member of the Judiciary Committee, said Wednesday that selecting Sandoval would not change his opposition to selecting a new justice this year.

"I don't know why Sandoval's name's come up, but it's the timing more than anything else. In the middle of the most irritating presidential contest in history, people are at each other's throats and now want to get us into what would be the most contentious Supreme Court battle in history," Hatch told CNN. "Let's do it next year. And whoever's president, maybe a Democratic president, is going to have that right to pick whoever they want."

But one Republican senator, who asked not to be named in order to speak candidly, called the decision to float Sandoval's name a "deft move."

Obama's aides said he spent last weekend delving into detailed packets about potential candidates. He was seen carrying a large black binder, divided into nine sections, as he returned to his residence Friday evening.

In his post, Obama made no indication of what ideology he was seeking in a Supreme Court nominee, and the White House insists he remains open to a spectrum of candidates.

But in recent days his allies have suggested Obama select a moderate who has gained support in the past from Republicans, even as it appears increasingly unlikely that any nominee will gain traction among GOP lawmakers.

Biden suggested in interviews last week Obama pick a "consensus candidate" and not the most "liberal jurist" he could muster. Without naming specific judges, he said there were plenty of names on the federal bench who had enjoyed broad support from Republicans during their confirmation processes.

Sen. Chris Coons, a Democrat on the Judiciary Committee who occupied Biden's former seat in the Senate, echoed his Delaware colleague's sentiments.

"This is not the moment for partisanship, this is the moment for the President to show that he is serious about having our institution function and to challenge the Republicans by putting forward a nominee who has already recently been unanimously confirmed by this Senate," Coons told CNN. "Thus pushing the point that we ought to get back to work and we ought to do our job."

Many of those names currently serve on the U.S. Court of Appeals for the D.C. Circuit, one of the nation's most important panels since it hears challenges to certain federal agencies.

While the court is sometimes regarded as a stepping-stone for judges to eventually serve on the Supreme Court, a former top adviser to Obama suggested Tuesday the President may avoid picking a name from that lower panel.

"Because those cases are critical cases and there are several of them before them right now, I think he'll look elsewhere for a nominee," David Axelrod, now a CNN senior political commentator, said on "The Situation Room."