WASHINGTON — President Trump was more personally involved in his campaign’s effort to obtain Democratic emails stolen by Russian operatives in 2016 than was previously known, phone records introduced in federal court on Wednesday suggested.

Federal prosecutors disclosed the calls at the start of the criminal trial of Roger J. Stone Jr., Mr. Trump’s longtime friend, who faces charges of lying to federal investigators about his efforts to contact WikiLeaks during the 2016 campaign. Russian intelligence officers had funneled tens of thousands of emails they stole from Democratic computers to WikiLeaks, which released them at critical points during the presidential race.

The records suggest that Mr. Trump spoke to Mr. Stone repeatedly during the summer of 2016, at a time when Mr. Stone was aggressively seeking to obtain the stolen emails from Julian Assange, the founder of WikiLeaks. The prosecutors noted that they did not know what Mr. Stone and Mr. Trump had discussed. But they stressed that the timing of their calls dovetailed with other key developments related to the theft and release of the Democratic emails.

The phone records are the first concrete suggestion that Mr. Trump may have had a direct role in his campaign’s effort to benefit from Russia’s hidden hand in the election. At the very least, the calls and other evidence underscored the eagerness of senior campaign officials and other Trump associates to reap the rewards of what the special counsel, Robert S. Mueller III, later called a sophisticated and alarming covert Russian influence operation.