Mr. Ellison has been the early favorite in the slow-forming party contest, winning support from liberal powers such as Mr. Sanders, Senator Elizabeth Warren of Massachusetts and the A.F.L.-C.I.O., in addition to Senator Chuck Schumer of New York, the incoming minority leader. After finding considerable unease among committee members over the prospect of electing a part-time chairman, Mr. Ellison said last week that he would resign his seat in Congress if he is elected.

Still, at party forums, in interviews and in private conversations, a range of Democrats have indicated an appetite for additional candidates to join the race, which also includes Jaime Harrison, the chairman of the South Carolina Democratic Party, and Raymond Buckley, the head of the New Hampshire Democratic Party. By heeding those calls, Mr. Perez is wagering that the committee members who will select the next party leader in late February are open to tapping somebody with few ties to the D.N.C.

While he has served in positions in state and federal government, Mr. Perez, 55, has scant experience with electoral politics, having served just four years on the Montgomery County Council in Maryland. The state party chiefs and other committee members who make up the voting members of the two national parties at times prefer candidates to come from their own ranks when they do not control the White House.

But there has been no groundswell of support from the committee for any of the announced candidates, and the contest remains highly volatile. In Mr. Perez, Democrats would have an articulate messenger who has ties to both the Hispanic community (his parents emigrated from the Dominican Republic) and the Rust Belt (he grew up in Buffalo).

He could potentially handle both the public and private aspects of the job, appearing on television as a face of the opposition to President-elect Donald J. Trump while also cultivating the party’s contributors. With many senior Democrats casting about for an alternative to Mr. Ellison — whose views on Israel have drawn scorn from Jewish groups and Haim Saban, one of the party’s most significant donors — Mr. Perez could prove a contender.