Cornyn's communications director confirmed the comments to Reuters.
Republican leaders have previously said they would not advance the nominations unless they were confident that Trump's controversial pick for one of the two open posts, former economic adviser Judy Shelton, has the support needed for approval.
But now they have another reason to advance the nomination: Republicans' control of the Senate, currently by 53 members, is only assured until January.
In the same elections earlier this month where Trump lost to Democrat Joe Biden, voters also gave Democrats one additional Senate seat. Control of two more seats - and of the Senate itself - hangs on the outcome of a Jan. 5 runoff in Georgia.
Critics have said Shelton could politicize the Fed's interest-rate setting because of her ties to Trump, though she told lawmakers in her confirmation hearing that "no one tells me what to do.
"The other Trump nominee, St. Louis Fed research director Christopher Waller, is less controversial.
Both would be regular voters on a 12-member rate-setting panel at the Fed, the world's most influential central bank.
Shelton's term would end in January 2024. Waller's would end in 2030.Two Republicans have said they will not support Shelton.