Mr. Richmond, the former head of the Congressional Black Caucus, said he believes that the former vice president is the party’s “best candidate to beat Trump.”

Still, in conversations over the phone and strategy sessions in his rented Northern Virginia home that are occasionally interrupted by his German shepherds, Major and Champ, Mr. Biden has acknowledged that he is uncertain about his place in the 2020 Democratic primary. He is also uneasy about potential attacks from rivals on his family, aides and advisers say, namely his son Hunter, a lawyer and lobbyist who went through a high-profile divorce in 2017 and has struggled with substance abuse.

And in a soliloquy that was unplanned if not unexpected to Biden advisers — rarely do the garrulous former senator’s private views remain private for very long — he openly gave voice to both concerns last week.

“I don’t think he’s likely to stop at anything, whomever he runs against,” Mr. Biden said about President Trump, expressing his alarm about putting his “family through what would be a very, very, very difficult campaign” without directly mentioning his younger son.

And then he mused about not wanting to pursue “a fool’s errand” in an appearance at the University of Delaware. “What I don’t want to do is take people’s time, effort and commitment without there being a clear shot that I could be the nominee,” Mr. Biden explained.

While he is leading the field, Mr. Biden is far from an overwhelming favorite and appears to have support from about a third of the Democratic electorate. Senator Bernie Sanders is close behind him. And even before he enters the race, the former vice president is facing questions about how his decades-long record — which includes support for the Iraq war, his role in enacting draconian sentencing guidelines and his skepticism toward Anita Hill during Justice Clarence Thomas’s Supreme Court hearings — will be received in today’s Democratic Party.