The White House is vetting Republican Gov. Brian Sandoval of Nevada for the Supreme Court seat left vacant by the late Justice Antonin Scalia, a source familiar with the process confirmed to CBS News on Wednesday.

Sandoval met with Senate Minority Leader Harry Reid, D-Nevada, on Monday while the governor was in Washington, D.C. for the National Governor's Association meeting, the source said.

The two spoke on the phone last week in Nevada and agreed to meet in person this week. The source said that they discussed the Supreme Court and that the White House was considering Sandoval.

The possible pick was first reported by The Washington Post on Wednesday.

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After the report surfaced, the White House would neither confirm nor deny that it was looking at the governor of Nevada for the nomination.

"I suspect that it is only the first of many stories that speculate on potential Supreme Court nominees and I don't think it would be helpful for me to get into a rhythm of responding to each one as it appears," spokesman Josh Earnest told reporters when he was asked whether Sandoval is a possibility.

Sandoval has served as Nevada's governor since 2011 and the Senate unanimously confirmed him in 2005 after President George W. Bush nominated him to be a judge for the U.S. District Court for the District of Nevada, which he served as until 2009.

It's unclear how many people the White House is eyeing, but President Obama was seen walking along the White House colonnade from the Oval Office last Friday evening carrying a thick binder of Supreme Court materials.

On Wednesday morning, Mr. Obama called on Senate Republicans to "fulfill their constitutional duties" and consider a high court nominee this year. He also penned a guest post for SCOTUSblog that laid out the type of person he would want to fill Scalia's seat.

"He or she will have an independent mind, rigorous intellect, impeccable credentials, and a record of excellence and integrity," the president wrote. "I'm looking for a mastery of the law, with an ability to hone in on the key issues before the Court, and provide clear answers to complex legal questions."

If Sandoval does wind up being the nominee, it would put Senate Republicans in a tough position. On Tuesday, their entire conference, including Senate Majority Leader Mitch McConnell, R-Kentucky and Judiciary Committee Chairman Chuck Grassley, R-Iowa, ruled out holding any confirmation hearings for anyone president appoints and said there would be no confirmation vote until a new president is sworn in next January.

"I believe the overwhelming view of the Republican conference of the Senate is that this nomination, this vacancy, should not be filled by this lame-duck president," McConnell said.

While the Republicans on the Judiciary Committee unanimously agreed that a nominee shouldn't be considered, there are a few other GOP senators who think the person should be given a chance.

"Should [Obama] decide to nominate someone to the Supreme Court, who knows, maybe it'll be a Nevadan," Sen. Dean Heller, R-Nevada, said in a statement last week.

This fight comes just days after Scalia was celebrated at a funeral mass last Saturday in Washington, D.C. and buried in a private ceremony. Scalia, 79, was found dead on Feb. 13 in his room at a West Texas resort.