While he did not discuss Mr. Trump’s role in his decision to resign at the time, he told the Senate panel that the president had been dismayed that Mr. Pak had investigated allegations of voter fraud in Fulton County, Ga., and not found evidence to support them, according to the person familiar with the statements.

Mr. Pak testified that top department officials had made clear that Mr. Trump intended to fire him over his refusal to say that the results in Georgia had been undermined by voter fraud, the person said. Resigning would pre-empt a public dismissal.

He also described work done by state officials and the F.B.I. to vet Mr. Trump’s claims of voter fraud, and said they had not found evidence to support those allegations.