WASHINGTON—President Donald Trump is actively discussing major changes in the White House, including having lawyers vet his tweets and shaking up his top staff, as he grapples with the fallout from probes into his campaign’s dealings with Russia, according to several senior administration officials and outside advisers.

Other revisions on the table include adding a roster of outside lawyers to help deal with the legal ramifications of the Russia investigation, officials and allies said. “Everything is in play,” one Trump...