WASHINGTON—President Donald Trump is expected to tell lawmakers as early as Friday that he has approved the release of a classified memo that Republicans allege shows improper surveillance of one of the president’s former campaign aides, a White House official said Thursday.

The official said Mr. Trump made his decision after reviewing the memo and ensuring “it doesn’t give away too much in terms of classification.” The official said the White House doesn’t expect the memo to include any redactions recommended by intelligence...