WASHINGTON—The White House is inclined to approve the release of a classified Democratic memo that rebuts Republican allegations of abuse by the FBI in its application to monitor a former Trump campaign aide, according to a person familiar with the matter.

John Kelly, White House chief of staff, said earlier this week that he told the lawyers reviewing the document to “get back to me” by the end of Thursday with their recommendation on whether President Donald Trump should release the memo. Mr. Kelly suggested then that redactions...