Lieberman was Gore's vice presidential running mate in the disputed 2000 election. He repeatedly has said he would not run if Gore does in 2004. "I think you've got to give Al Gore some room to make a very personal decision as to whether he and (his wife) Tipper want to do this again," Lieberman said.
A Los Angeles Times poll of 312 Democratic National Committee members published Sunday found about 20 percent favored Gore, while 4 percent backed Lieberman. Choosing from a list of 10 prospective candidates, almost half of respondents had no preference.
Among other Democrats considering running for president in 2004 are outgoing House Minority Leader Dick Gephardt of Missouri, Senate Majority Leader Tom Daschle of South Dakota, Massachusetts Sen. John Kerry, Vermont Gov. Howard Dean and North Carolina Sen. John Edwards.