Starship is SpaceX's next-generation rocket crucial for the company's commercial launch business and Musk's aim to start human colonies on Mars.
The U.S. offers few such options and export controls would make building a foreign launch site difficult.
SpaceX has eyed another Kennedy Space Center launch site for future Starship launches, LC-49, a few miles from LC-39A.
But that location is in the midst of a lengthy environmental review that could take years.
Plans for that orbital launch site, Spaceport Camden, were nixed by a local referendum after a lawsuit raised concerns about its environmental impact.