WASHINGTON—Lawyers for Donald Trump have met several times with special counsel Robert Mueller in recent months and submitted memos arguing that the president didn’t obstruct justice by firing former FBI chief James Comey and calling into question Mr. Comey’s reliability as a potential witness, people familiar with the matter said.

The legal arguments and meetings offer the first detailed look at the interplay between the wide-ranging investigation and the team that is representing the president since the special counsel was...