Also under scrutiny is how forthcoming Mr. Kushner was with his father-in-law about the nature of the June meeting. He met with Mr. Trump to discuss the issue, according to advisers to the White House, around the time he updated his federal disclosure form to include Ms. Veselnitskaya’s name on a list of foreign contacts that Mr. Kushner was required to submit to the F.B.I. to obtain a security clearance.

Mr. Kushner supplemented the list of foreign contacts three times, adding more than 100 names, people close to him said.

Mr. Kushner played down the significance of the meeting and omitted significant details, according to two people who were briefed on the exchange. They said Mr. Kushner informed the president that he had met with a Russian foreign national, and that while he had to report the name, it would not cause a problem for the administration.