Sally Q. Yates, the deputy attorney general in the final years of the Obama administration, had long been considered the front-runner. Mr. Biden is close to her and has told friends that he could imagine her as the nation’s top law enforcement official. But some advisers fear that Republicans would block her nomination because of her refusal to defend Mr. Trump’s first travel ban and her role in the early stages of the investigations into his campaign and associates.

Mr. Biden could instead pick Lisa Monaco, the former homeland security adviser for President Barack Obama who was a finalist in 2013 to be F.B.I. director. And like Ms. Yates, she worked well with Mr. Biden when he was vice president…

But both women are up against Deval Patrick, the former Massachusetts governor who served as the head of the department’s civil rights division in the Clinton administration and would become the second Black man to be attorney general…

Others around the president-elect are not eager to reward Mr. Patrick, who jumped into the Democratic nomination last year to challenge Mr. Biden as a politically moderate answer to the party’s more liberal candidates. Xavier Becerra, the attorney general of California, is also under consideration for attorney general.