President Biden is nearing a final decision to formally enter the 2024 presidential race as early as Tuesday, with a video to announce his run already in production, according to four people with knowledge of the plans.

Mr. Biden, who said last week while in Ireland that he would enter the race “relatively soon,” will spend the weekend at Camp David, and he is expected to be joined by family members and some advisers. He has not yet given final approval to the announcement plan, according to one person with knowledge of the discussions.

The New York Times reported on Monday that the Biden operation was discussing the possibility of a low-key video announcement next week on Tuesday, which marks the fourth anniversary of his entry into the 2020 race. One of Mr. Biden’s favorite poems, which he has often quoted, is about making “history and hope rhyme.”

On Thursday, The Washington Post first reported that plans for an announcement next week were being finalized with Tuesday as a target.

Karine Jean-Pierre, the White House press secretary, declined to comment in a text Thursday evening. At a press briefing earlier in the day, she told reporters: “What I will say is that any announcement or anything that is related to 2024 certainly will not come from here.” The Democratic National Committee did not respond to a request for comment.

At 80, Mr. Biden is already the oldest president in American history and, by the end of a potential second term, he would be 86.

The timing of a 2024 decision has been closely held by Mr. Biden’s inner circle at the White House, where re-election planning has been underway for months, overseen by two top advisers, Jennifer O’Malley Dillon and Anita Dunn. Still, planning has intensified in recent weeks with meetings between White House advisers and Democratic Party officials, with a focus on what kind of apparatus would support the president from the outside.

Mr. Biden has a long history of extending deadlines around making major political decisions, injecting a measure of uncertainty into the timetable in the eyes of some of his allies.

The political durability of the Republican front-runner, former President Donald J. Trump, has added to Mr. Biden’s ability to keep a coalition of Democrats together, including progressives who have at times taken issue with some White House policies. Mr. Trump, who has continued leading polls despite being indicted by the Manhattan district attorney this month, has proved to be a glue holding factions of the Democratic Party in place since 2020, when Mr. Biden won the South Carolina primary after losing the first two early state contests.

Mr. Biden has already summoned donors to Washington next week, inviting those who have given at least $1 million to a two-day gathering starting on Friday. The event, which is not a fund-raiser, is intended to rally his army of bundlers and donors ahead of a 2024 campaign that is likely to top more than $1 billion, including super PAC spending.

Cash considerations have been at the center of the Biden team’s thinking for when to enter the race. Announcing will allow him to begin banking contributions from big and small donors, but opening a campaign will incur significant expenses that might otherwise be deferred.

Some outside groups have already begun preparing for a campaign, including a group called Future Forward that is expected to take the lead in television advertising; the long-running Democratic super PAC Priorities USA, which primarily focuses on digital work; and the group American Bridge, which has held events attended by administration officials.

For instance, Ms. Dunn attended an American Bridge conference in Fort Lauderdale and appeared as a keynote speaker in her personal capacity.

Mr. Biden is expected to face only token opposition in the primary. The author Marianne Williamson, who ran and lost in 2020, and the anti-vaccine activist Robert F. Kennedy Jr. are both running long-shot campaigns.

Chris Cameron and Katie Glueck contributed reporting.