Mr. Biden, a veteran of the Senate who for decades has believed in negotiating on the particulars of voting rights legislation, has faced calls to push Democratic senators to eliminate the filibuster, which would allow the two major voting bills proposed by the party to pass with a simple majority. The president and his advisers have repeatedly pointed out that he does not have the votes within his own party to pass federal voting legislation, and does not have the power to unilaterally roll back the filibuster even if he supported doing so.

But voting rights groups say that Mr. Biden is not expending sufficient political capital or using the full force of his bully pulpit to persuade Congress. They point to the contrast between his soaring language — “Jim Crow on steroids,” he has called the G.O.P. voting laws — and his opposition to abolishing the Senate filibuster.

“As you noted in your speech, our democracy is in peril,” the groups said in their letter. “We certainly cannot allow an arcane Senate procedural rule to derail efforts that a majority of Americans support.”