Ms. Bass has also waged a previously undisclosed campaign to woo influential liberal leaders, telephoning union presidents to seek their counsel and support.House Speaker Nancy Pelosi is close with Ms. Bass, whom she named to oversee the recent policing reform bill, and has made her admiration clear in private conversations, including with former President Barack Obama. Ms. Pelosi’s aides said she has not conveyed support for any one candidate, is fond of a number of them and, in speaking with Mr. Biden’s vetting team last month, urged them to find somebody who could ensure the ticket is victorious.Two prominent Democrats, Senators Elizabeth Warren of Massachusetts and Kamala Harris of California, along with a handful of other women, remain as possibilities; both lawmakers have the statewide political experience and large national followings that Ms. Bass and Ms. Rice lack. Ms. Warren has become something of an informal adviser to Mr. Biden on economic issues and has won support from her party’s progressive wing, and Ms. Harris is regarded as a muscular fund-raiser with the backing of important people in the Democratic Party’s donor class.While Mr. Biden’s advisers are careful to stress that he has not ruled out any of the major candidates, some are clearly less likely than others to be chosen. For some of the long-shot candidates, talk has already turned to other potential roles in a Biden administration: Gov. Michelle Lujan Grisham of New Mexico, for instance, has expressed interest in the job of health and human services secretary, according to officials familiar with her thinking.Among the other candidates Mr. Biden has looked at closely are Gov. Gretchen Whitmer of Michigan; Representative Val Demings of Florida, who has enlisted her home state’s sizable congressional delegation to make appeals on her behalf; and Senator Tammy Duckworth of Illinois, who is backed by veterans advocates.Gov. J.B. Pritzker of Illinois has called Mr. Biden’s team to urge them to put Ms. Duckworth, a military veteran, on the ticket, and Senator Jack Reed of Rhode Island, the top Democrat on the Senate Armed Services Committee, has shared his high opinion of the combat-wounded Ms. Duckworth with the Biden camp, and also urged them to pick his home-state governor, Gina Raimondo, people familiar with the conversations said.