Several Republican senators have taken steps to consider Merrick Garland’s nomination to the Supreme Court despite calls from their leadership to leave the selection to the next president.
Republican leader Mitch McConnell has said the Senate will not hold confirmation hearings.
However, some Republican senators in tough re-election battles have agreed to meet Judge Garland.
One of them, Mark Kirk of Illinois, will meet Mr Garland this week.
President Barack Obama nominated Mr Garland, a veteran federal appeals court judge, to fill the vacancy left by the death of Justice Antonin Scalia.
If Mr Garland is confirmed, it would shift the balance of power on the court, which often has the final say on divisive issues in US politics.
Appointed to life time terms, justices frequently face a contentious confirmation process.
Scalia had been part of a conservative majority on the court for several years.
Republican leaders maintain that court vacancies should not be filled in an election year. However, no formal rules prohibit justices from being confirmed prior to elections.
Several national polls have found a majority of voters – particularly independents – favour holding hearings for Mr Garland.
Senators Pat Toomey of Pennsylvania and Kelly Ayotte of New Hampshire – both facing formidable opponents in November – have also agreed to meet Mr Garland.
However, Ron Johnson of Wisconsin and Rob Portman of Ohio – also locked in tight re-election fights – have stood firm, despite pressure from their opponents and Democratic groups.
The meetings, a traditional courtesy for court nominees, do not mean the senators support hearings for Mr Garland, but some conservative critics see them as a first step towards hearings.
Some Republicans already support hearings.
Senator Jerry Moran of Kansas, moderate Republican Senator Susan Collins of Maine and Senator Kirk have recently broken with party leaders and called for hearings.
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Mr Garland is seen as more moderate than Mr Obama’s previous picks for the court and several sitting Republican senators supported him when he was selected for his current post in 1997.
At least seven Republican senators would have to join Democrats to overcome procedural hurdles and confirm Judge Garland.
Privately, some Republicans have said they would support Mr Garland’s nomination during the so-called “lame-duck” session after the presidential election in November.
Many Republicans have acknowledged that Mr Garland would be preferable to a less moderate nominee selected by either Hillary Clinton or Bernie Sanders.
Others worry that current Republican front-runner Donald Trump is too unpredictable to be trusted with a court nomination.