WASHINGTON—The White House denied a request to release a Democratic memo that aimed to rebut GOP allegations regarding the investigation of a onetime associate of President Donald Trump, sending it back for revisions due to national-security concerns.

“Although the president is inclined to declassify the February 5th memorandum, because the memorandum contains numerous properly classified and especially sensitive passages, he is unable to do so at this time,” White House counsel Donald McGahn wrote to the House Intelligence...