Six weeks after Election Day, with his hold on power slipping, President Donald J. Trump directed his lawyer, Rudolph W. Giuliani, to make a remarkable call. Mr. Trump wanted him to ask the Department of Homeland Security if it could legally take control of voting machines in key swing states, three people familiar with the matter said.

Mr. Giuliani did so, calling the department’s acting deputy secretary, who said he lacked the authority to audit or impound the machines.

Mr. Trump pressed Mr. Giuliani to make that inquiry after rejecting a separate effort by his outside advisers to have the Pentagon take control of the machines. And the outreach to the Department of Homeland Security came not long after Mr. Trump, in an Oval Office meeting with Attorney General William P. Barr, raised the possibility of whether the Justice Department could seize the machines, a previously undisclosed suggestion that Mr. Barr immediately shot down.

The new accounts show that Mr. Trump was more directly involved than previously known in exploring proposals to use his national security agencies to seize voting machines as he grasped unsuccessfully for evidence of fraud that would help him reverse his defeat in the 2020 election, according to people familiar with the episodes.