WASHINGTON—President Donald Trump met Tuesday with Deputy Attorney General Rod Rosenstein to discuss a classified Democratic memo rebutting GOP allegations of partisan motives in the federal investigation of his former campaign aide, as the White House decides whether to authorize the document’s release.

White House press secretary Sarah Huckabee Sanders said Tuesday that the president met with Mr. Rosenstein “to discuss some of the differences” between the Democratic memo and the Republican document it addresses, which was...