The president of the Teamsters union has asked for speaking slots at both the Republican and Democratic national conventions, at a time when President Biden and former President Donald J. Trump have pressed for support from rank-and-file members of organized labor.

The move by Sean O’Brien, the president of the International Brotherhood of Teamsters, underscores the fact that his group, unlike other influential umbrella unions that have backed Mr. Biden in the 2024 election, has yet to endorse in the presidential race. Mr. O’Brien has made clear he is delaying a decision until later this year.

Kara Deniz, a spokeswoman for the Teamsters union, confirmed that Mr. O’Brien, through aides, has told officials working on both conventions that he would be interested in speaking at their dayslong nomination events. The Republicans will hold their convention in Milwaukee in July, and the Democrats will hold theirs in Chicago in August.

It would be unusual in the current fractious political climate for someone to speak at both conventions.

Over the course of the year, Mr. O’Brien has invited several presidential candidates, including Mr. Biden, Mr. Trump and independent candidates like Cornel West and Robert F. Kennedy Jr., to speak before his group. But Mr. O’Brien has what people close to Mr. Trump believe is a developing relationship with the former president.

The Teamsters is one of the country’s largest labor unions, with 1.3 million members in sectors like trucking and manufacturing. The A.F.L.-C.I.O. and the United Automobile Workers have backed Mr. Biden, and Shawn Fain, the president of the U.A.W., has been harshly critical of Mr. Trump.