Mr. Biden, after presiding over the Senate Monday night, told a small group of reporters at the Capitol that “I am going to run in 2020,” according to NBC News. Asked what he will run for, the vice president replied, “for president, and also, what the hell man, anyway.”

Offered a chance to say he was joking, Mr. Biden did not take it.

“I’m not committing not to run,” he said. “I’m not committing to anything. I learned a long time ago, fate has a strange way of intervening.”

Based on those remarks, Jon Cooper, who was national finance chairman for last year’s Draft Biden effort to coax the vice president into the 2016 presidential campaign, purchased a series of web domains including draftbiden2020.com, biden2020.net and runjoerun2020.com.

Mr. Cooper said he had no information beyond Mr. Biden’s remarks at the Capitol to suggest the vice president, who will be 77 years old by Election Day 2020, is weighing a third attempt at the White House. Mr. Biden lost in the 1988 and 2008 Democratic presidential primaries before President Barack Obama tapped him as his running mate.

“I have no inside info whatsoever,” Mr. Cooper said, declaring himself “an eternal optimist.”

No other Democrat has so openly suggested they are considering running for president in four years, when President-elect Donald Trump would be running for re-election should he seek a second term.

Mr. Biden considered seeking the presidency in 2016, but opted out of a campaign after months of deliberations following his son Beau’s death. Announcing he wouldn’t run in October 2015, Mr. Biden said it was too late to mount a winning campaign.

“Unfortunately, I believe we’re out of time, the time necessary to mount a winning campaign for the nomination,” he said.

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