WASHINGTON—President Trump said again on Friday that he had no prior knowledge of a June 2016 meeting that his eldest son had with a Russian lawyer at Trump Tower after being promised damaging information about Democratic presidential candidate Hillary Clinton.

The president disputed an account that a person familiar with the matter says his former lawyer Michael Cohen is planning to tell the team of Robert Mueller, the special counsel investigating alleged Russian interference in the 2016 election.

In a series of tweets Friday morning, Mr. Trump wrote, “So the Fake News doesn’t waste my time with dumb questions, NO, I did NOT know of the meeting with my son, Don jr.”

Mr. Trump also took a shot at Mr. Cohen’s lawyer: “He even retained Bill and Crooked Hillary’s lawyer. Gee, I wonder if they helped him make the choice!”

Lanny Davis, who is Mr. Cohen’s lawyer and served as a special counsel to President Clinton in the 1990s, declined to comment on the tweet.