Former Vice President Joseph R. Biden Jr. is under pressure from Democratic donors and others to make up his mind about a 2020 presidential run by the end of January. Rick Bowmer/Associated Press

WASHINGTON — Former Vice President Joseph R. Biden Jr. is in the final stages of deciding whether to run for president and has told allies he is skeptical the other Democrats eyeing the White House can defeat President Trump, an assessment that foreshadows a clash between the veteran Washington insider and the more liberal and fresh-faced contenders for the party’s 2020 nomination.

Many Democratic voters, and nearly all major Democratic donors, are keenly interested in Mr. Biden’s plans because of their consuming focus on finding a candidate who can beat a president they believe represents a threat to American democracy. But there is also a rising demand in the party for a more progressive standard-bearer who reflects the increasingly diverse Democratic coalition.

Mr. Biden would instantly be the early front-runner if he ran, but he would have to bridge divides in a primary that would test whether Democrats are willing to embrace a moderate white man in his 70s if they view him as the best bet to oust Mr. Trump.

“He has the best chance of beating Trump, hands down,” said Senator Tom Carper of Delaware, Mr. Biden’s longtime friend and former colleague. “On a scale of one to 10, that’s probably about a 12 for us.”

Yet Mr. Biden’s skepticism about the field could alienate female and minority voters who are excited that several women and African-Americans are expected to run. Nominating a white man may also roil some Democrats who are already torn about whether a woman could win in 2020 after Hillary Clinton’s loss. And Mr. Biden’s preoccupation with winning back blue-collar Midwestern whites could place him at odds with Democrats who see greater potential for growth in the highly educated suburbs and across the booming Sun Belt and upper South.

More broadly, debate around Mr. Biden’s possible candidacy illustrates the dueling visions in the party and particularly the divisions between its pragmatic and liberal wing. Some Democrats are skeptical that a relatively moderate candidate like Mr. Biden, who has baggage like supporting the 1990s crime bill that is loathed on the left, would prevail in the primary with a message of unity and national healing rather than the fiery and uncompromising brand of populism that Democratic primary voters elevated in the midterm elections.

“In 2020, Biden-style centrism will become a toxic and losing brand of politics in Democratic primaries,” said Waleed Shahid, a left-wing activist.

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The 76-year-old former vice president, who leads the field in initial national and Iowa polls, has not yet told his allies that he has decided to run. And they emphasize that Mr. Biden’s decision will not be final until he says it for himself, noting that family considerations are central. Mr. Biden sought the presidency twice before but he has also considered it at least as many times before bowing out.

But Mr. Biden has indicated that he is leaning toward running and will most likely make a decision within the next two weeks, according to Democrats within and beyond his inner circle who have spoken to him recently.

The former vice president told a senior Democratic official last week that he is both likely to run and that his aides have told him he must move quickly in this primary, according to two Democrats briefed on the conversation.

Mr. Biden is under significant pressure from his party’s major donors to make up his mind — no later than the end of January — or risk watching them migrate to other candidates. Mr. Biden and his advisers are highly sensitive to the criticism that in 2015, he dragged out his decision before ultimately opting out of the Democratic primary, exasperating his supporters and undermining the Democratic front-runner, Mrs. Clinton, in the process.

Senator Elizabeth Warren of Massachusetts entered the Democratic primary race last week, and several other women are considering a 2020 run as well.

He spent some of his Christmas vacation in the Virgin Islands calling and texting his would-be supporters and is using this week to hold private meetings in Washington with Democratic officials.

In one of his calls over the holidays, Mr. Biden repeated a variation of a line he has used publicly: “If you can persuade me there is somebody better who can win, I’m…