If the White House was hoping to avoid bringing attention to ongoing Washington investigations into whether Mr. Trump’s associates colluded with Moscow, the president’s tweet thrust the saga back into spotlight. It was the latest example of his history of writing incendiary tweets that seize attention.

American intelligence officials have said that Mr. Podesta’s emails were hacked and stolen by the Russians and released to help Mr. Trump win the presidency. The D.N.C.’s emails were also stolen in a separate intrusion during the 2016 campaign.

The F.B.I. has been investigating the hacks and possible coordination between Russia and Mr. Trump’s associates since last summer, a probe that has dogged his presidency and, in doing so, infuriated the commander-in-chief.

Mr. Trump has been reluctant to accept Russia’s role in the hacks, and on Thursday said “nobody really knows for sure” if Moscow was involved. And some Republicans have focused on the D.N.C.’s refusal to share its computer server with the F.B.I. In another tweet on Friday, Mr. Podesta reminded Mr. Trump that he had nothing to do with the D.N.C.