WASHINGTON—Sen. Lindsey Graham, set to take over the Senate Judiciary Committee in January, has become an increasingly outspoken ally of President Trump in recent months, creating a potentially sharp split between the chambers on legal matters as Democrats take over the House.

Sen. Chuck Grassley, an Iowa Republican, said late last week he would leave his role as Judiciary chairman to take the helm of the Finance Committee. That clears the way for Mr. Graham, a South Carolina Republican, to run one of the Senate’s most powerful...