A lawyer who tried to help Donald J. Trump overturn his 2020 presidential election loss has been in direct contact with Mr. Trump once again as his allies start to lay the groundwork for challenging this year’s election results in key battleground states, according to two people with knowledge of their discussions.

The lawyer, Kurt Olsen, has spoken to Mr. Trump multiple times in recent weeks, the people said. In 2021, Mr. Olsen spoke to Mr. Trump several times by phone on the day of the Jan. 6 attack on the Capitol by a pro-Trump mob, and a federal judge later imposed sanctions on him for filing baseless claims about the midterm elections in Arizona in 2022.

Mr. Olsen has insisted to Mr. Trump that it’s important to make legal demands before Election Day related to the preservation of data from voting machines in Arizona, Wisconsin and Georgia, according to one of the people with knowledge of the discussions.

The former president is said to have been generally encouraging of Mr. Olsen. Others in Mr. Trump’s orbit view the continuing demands by lawyers like Mr. Olsen about the voting machines as potentially detrimental to any valid legal efforts Republicans might file should there be contested elections in the battleground states. Those Republicans believe judges will view such demands as frivolous.

Mr. Olsen did not immediately respond to multiple requests for comment. A Trump campaign spokesman did not respond to an email seeking comment.

In late 2020, Mr. Olsen, a lawyer based in Washington, D.C., got to know Mr. Trump as the former president listened to almost anyone offering to help him stay in power. Back then, Mr. Olsen was a little-known lawyer with a history of defending companies in product liability cases. He had joined the Texas attorney general’s failed effort to persuade the Supreme Court to stop four battleground states from certifying Joseph R. Biden Jr.’s victory over Mr. Trump.