Lee is best known for playing the title character on the sit-com "My Name is Earl." He has also starred in the movies "Chasing Amy," "Dogma" and "Vanilla Sky."
TMZ.com said he and Alkac were spotted in line for a marriage license at the Norwalk, Calif. courthouse Monday afternoon.
Lee and Alkac are expecting their first child to be born in the fall.
The actor also has a son named Pilot Inspektor from a previous relationship.
Calls to Lee's representatives requesting a comment on the story weren't returned, TMZ.com said.