President Trump’s eldest son, Donald Trump Jr., was promised damaging information about Hillary Clinton before agreeing to meet with a Kremlin-connected Russian lawyer during the 2016 campaign, according to three advisers to the White House briefed on the meeting and two others with knowledge of it.

The meeting was also attended by the president’s campaign chairman at the time, Paul J. Manafort, as well as by the president’s son-in-law, Jared Kushner. Mr. Manafort and Mr. Kushner recently disclosed the meeting, though not its content, in confidential government documents described to The New York Times.

The Times reported the existence of the meeting on Saturday. But in subsequent interviews, the advisers and others revealed the motivation behind it.

The meeting — at Trump Tower on June 9, 2016, two weeks after Donald J. Trump clinched the Republican nomination — points to the central question in federal investigations of the Kremlin’s meddling in the presidential election: whether the Trump campaign colluded with the Russians. The accounts of the meeting represent the first public indication that at least some in the campaign were willing to accept Russian help.