The plane took off from Clarksdale, Mississippi, on Saturday evening.
Its tires ruptured after takeoff, forcing it to land at Tunica Airport in Mississippi, attorney William Luckett said.
The plane skid off the runway as it landed, damaging it further, according to Luckett, who is also the mayor of Clarksdale.
Another aircraft was scheduled to pick Morgan up and take him to Houston.
A law enforcement source said authorities received a 911 call following the incident. Two people were aboard the plane, but no injuries were reported, the source said.
The Oscar-winning actor is known for his authoritative voice and calm demeanor.
He has appeared in various movies, including “Million Dollar Baby,” “The Shawshank Redemption” and “Invictus.”