Following Mr. Trump’s wishes, his advisers have aggressively pushed the Clinton camp emails in news media briefings and cable news appearances, bringing up the hacked messages to battle back from the questions about Mr. Trump’s comments about women. But as much as Mr. Trump sees WikiLeaks coming to his rescue, strategists in his own party take a dim view of its ultimate impact.

The Clinton campaign is trying its own political jujitsu with the hacks, arguing that they are more evidence that Mr. Trump is in the pocket of Mr. Putin, whom the Republican candidate has declined to denounce for his annexation of Crimea, his intimidation of former Soviet states that are now part of NATO, or for its abandonment last week of a nuclear arrangement with the United States. Mr. Podesta has gone even further, saying in a statement on Wednesday evening that there was “the possibility that Trump’s allies had advance knowledge of the release of these illegally obtained emails.”

Intelligence officials say that so far they have not concluded there was any such collusion, but the investigation into who got into Mr. Podesta’s emails, and how they got into WikiLeaks’ hands, has just begun.

Those emails began to appear on Friday afternoon, just hours after the director of National Intelligence and the Department of Homeland Security issued a statement attributing previous hacks to the Russian government. Officials on Wednesday said it may take weeks to establish whether Mr. Podesta’s emails were also hacked by the Russians — though they said the attack on his Gmail account fits the pattern of previous Russian-sponsored email thefts.

Republicans have previously condemned WikiLeaks and similarly blasted the leaks by Edward J. Snowden, a National Security Agency contractor, and said they were evidence of carelessness by the Obama administration.

When Mr. Snowden’s disclosures about the scope of the N.S.A. spying were brought to light, it touched off a feverish debate over government invading people’s privacy, and many Republicans denounced Mr. Snowden as a traitor. The emails from Mr. Podesta were also the result of an illegal hack — but of a private email account or campaign emails, not a government agency.