Israeli financial daily Globes reported Mr Musk is considering an investment or acquiring the company outright.
The South African-born, US-based billionaire also took time out to see the sites in Israel, posting pictures for his 6.8 million Instagram followers.
He snapped a selfie at Masada saying: "Paid respects to Masada earlier today. Live free or die."
He later poured flaming absinthe over a tower of glasses in Jerusalem bar Gatsby and drank at O'Connell's Irish bar in the Mahanei Yehuda market.
This week's visit is not Mr Musk's first to the country.
He was in Jerusalem in 2016 to visit Mobileye, a developer of sensors for autonomous cars that were briefly installed in Tesla vehicles, although the collaboration ended after six months amid mutual recriminations following a car accident.
But Mobileye appeared to suffer no long-term harm because it was acquired by Intel for $15.3 billion last year.
Nor did the episode, it seems, dampen Mr Musk's appetite for Israeli technology.
