This comes after Musk tweeted his frustration with California.
In a letter written to Musk, Turner touted Houston as the energy capital of the world and home to 22 Fortune 500 companies.
On Saturday, Musk threatened to pull the company's factory and headquarters out of California in an escalating spat with local officials who have stopped the company from reopening its electric vehicle factory.
READ MORE: Elon Musk threatens to relocate Tesla over California coronavirus restrictions
"Our city's long-standing history of industrial innovation and corporate collaboration offers unmatched advantages for production and growth and our large consumer market provides direct access to a valuable customer base," said Turner.
Turner closed the letter by saying, "I welcome the opportunity to further discuss the benefits of consolidating your companies in our region and more formally introduce you to the benefits of doing business in our great city."