Vice President Joe Biden’s aides in recent days called Democratic donors and supporters to suggest he is more likely than not to enter the 2016 race, and their discussions have shifted toward the timing of an announcement, said people familiar with the matter.

While the Biden team is still debating the best time to jump in, the vice president met Monday with his political advisers and talked about the merits of an early entry that would assure him a place in the Democratic debate scheduled for Oct. 13. They also are honing his campaign message and moving ahead with plans to raise money and hire staff, the people said.

The vice president could still back out if he finds the emotional toll from the death of his elder son, Beau Biden, too much to overcome. But the shift in the deliberations to the timing of an announcement is a fresh sign that he is moving closer to a bid.

“It’s my sense that this is happening, unless they change their minds,” said one person who spoke to Biden aides last weekend.

Mr. Biden’s entry would coincide with Democratic front-runner Hillary Clinton’s ramped-up efforts to reassure her backers that the probes into her use of a personal email server while she was secretary of state won’t derail her candidacy. That controversy has produced a month of bad headlines for Mrs. Clinton and helped boost her chief rival, Vermont Sen. Bernie Sanders, in the polls. A Biden bid could make her road to the Democratic nomination even tougher.