Joe Biden has told Barack Obama he will run for re-election in 2024, according to a Washington website, the Hill.

The site cited two anonymous sources. One was quoted as saying Biden “wants to run and he’s clearly letting everyone know”.

It was not clear when Biden told Obama his plans. But Obama visited the White House earlier this month, to celebrate the Affordable Care Act.

Introducing his host, Obama called him “Vice-President Biden”.

“That was a joke,” he said, to laughter.

Biden was vice-president to Obama from 2009 to 2017. He won the presidency on his own third attempt in 2020 (after short-lived campaigns in 1988 and 2008), beating Donald Trump.

At 78, Biden was the oldest president ever inaugurated. If he won again he would be 82 at the start of his second term.

Amid competing crises, from the Russian invasion of Ukraine to the coronavirus and rampant inflation at home, Biden’s approval ratings have plummeted. Most observers expect Republicans to retake Congress in November. But the president has dropped plenty of hints that he does plan to run again.

In September, it was widely reported that Biden and aides had told allies he planned to run again. In December, Biden said he would run if he stayed in good health. He has also said Kamala Harris, his vice-president, would be on the ticket again.

In his December interview with ABC News, Biden said another Trump candidacy would “increase the prospect of running”.

In Brussels in March, Biden referred to Trump again when he said: “In the next election, I’d be very fortunate if I had that same man running against me.”

Trump, 75, is readying his own third run for the presidency.

One of the sources who spoke to the Hill said Biden “thinks he’s the only one who can beat Trump. I don’t think he thinks there’s anyone in the Democratic party who can beat Trump and that’s the biggest factor.”

In 2020, Biden surged to victory in the Democratic primary on the back of support from Black voters – and a quickly gathering sense that he was indeed the party’s best hope of beating Trump.

According to books and reportage about the 2020 campaign, Obama doubted whether Biden could win.

Edward-Isaac Dovere, author of Battle for the Soul: Inside the Democrats’ Campaigns to Defeat Trump, reports: “Obama was doubtful about [Biden’s] prospects. He didn’t think Biden could be a disciplined enough candidate.”

Dovere also says Obama doubted Biden’s stamina for the race, was not sure he had the requisite “swagger” for an American president, and worried his vice-president had “trouble … connecting with crowds”.

But Dovere also quotes Jen Psaki, now Biden’s White House press secretary, as saying Obama “undervalued Biden’s political abilities because they had such different styles”.