WASHINGTON — President Obama is vetting Merrick B. Garland and Sri Srinivasan, federal appellate judges who have enjoyed substantial support from Republicans in the past, as potential nominees for a Supreme Court vacancy that has set off a brutal election-year fight.

Judge Garland, 63, and Judge Srinivasan, 49, are undergoing background checks by the F.B.I., according to a person knowledgeable about the process, who spoke on the condition of anonymity because it has been cloaked in secrecy.

The White House is also vetting Ketanji Brown Jackson, 45, a federal trial judge, according to The National Law Journal, which cited a lawyer who had been interviewed by the F.B.I. Judge Jackson formerly served on the United States Sentencing Commission and is related by marriage to the House speaker, Paul D. Ryan, Republican of Wisconsin. Earlier this week, the president was reported to also be vetting Jane L. Kelly, another federal appeals court judge.