A number of bankers and private-equity investors privately cast doubt on Elon Musk’s ability to pull off a buyout of Tesla Inc. after he raised the possibility in a series of tweets Tuesday.

The electric-car company’s chief said in one tweet that as part of the deal he envisions, Tesla’s current investors could either sell or keep their stakes. He said he would create a “special purpose fund enabling anyone to stay with Tesla,” likening the prospect to an arrangement he said his Space Exploration Technologies Corp. has with Fidelity...