Some lawmakers are also complaining that they have not been adequately consulted by Mr. Biden and his team on appointments, especially the selection of Lloyd J. Austin III, a retired general, for defense secretary, given that his recent military service will require Congress to grant him a special waiver.

Identity-based groups continue to lobby Mr. Biden to ensure racial and gender diversity at all levels of his administration. Last week, Hispanic lawmakers directly pressed Ms. Harris to ensure there would be at least two Latinas in prominent administration roles — a commitment she declined to make, according to people briefed on the meeting.

“She didn’t commit,” said Representative Vicente Gonzalez of Texas, “but she did commit to work with us regularly.”

That commitment was not enough, however: On Monday, a group of Latina Democrats in Congress called for Mr. Biden to nominate at least two Latinas to his cabinet. Also on Monday, a constellation of Hispanic advocacy groups had a Zoom meeting with an incoming White House aide, Cedric Richmond, and pressed him to make Mr. Biden available for a meeting similar to the one with civil rights leaders last week.