Even before announcing a candidacy, Mr. Biden has started trying to rebut those concerns, telling party officials in Delaware this month that he has “the most progressive record” of anyone running for president.

But the issue of abortion poses particularly challenging terrain for Mr. Biden. Efforts to restrict access to abortion by the Trump administration, and the new conservative majority on the Supreme Court, have heightened concerns among many Democrats that federal protections of abortion rights could be chipped away or eventually overturned — and that the next president needs to be a dependable ally on abortion issues.

“Anxiety is super high among women across the country,” said Ilyse Hogue, president of the abortion rights organization Naral Pro-Choice America. “Joe Biden is trying to carve out a space for himself as the middle, moderate candidate, and he’s going to have to really get with the times and understand that standing with abortion rights is the middle, moderate position.”

She added, “I can’t tell you if he’s there or not.”

Mr. Biden is already facing criticism from some women’s rights activists over his aggressive questioning of Anita Hill during the Supreme Court confirmation hearings of Judge Clarence Thomas in 1991. Mr. Biden’s comment Tuesday that he wished he “could have done something” to give Ms. Hill’s claims of sexual harassment a more respectful hearing drew fierce backlash from critics, who pointed out that Mr. Biden was chairman of the Senate committee that questioned Ms. Hill. Some women’s rights leaders say Mr. Biden must offer a stronger and more personal apology to Ms. Hill, as well as clarify his views on a broad range of issues including sexual assault, harassment and Republicans’ efforts to limit abortion access. (Mr. Biden has spoken warmly about some Republicans and bipartisanship in recent months.)

[Read Michelle Goldberg’s column about how the abortion divide is getting deeper in states.]

Mr. Biden declined to be interviewed for this article. His spokesman, Bill Russo, said the former vice president is a supporter of the Roe decision who fought to protect abortion rights by mounting a fierce opposition to the nomination of a conservative judge, Robert H. Bork, to the Supreme Court in 1987.

“Because of that, Roe and its progeny have been preserved for 30 years. But for that effort, Roe v. Wade would not be the law of the land today,” Mr. Russo said.