Billionaire investor Kirk Kerkorian says he may buy millions more shares in General Motors Corp., as the mogul ratchets up the pressure on the auto maker to explore an alliance with Renault SA and Nissan Motor Co. Ltd.

Kerkorian’s investment company, Tracinda Corp., already owns 56 million shares in GM, or 9.9% of the company’s outstanding stock.

In a letter dated today, Tracinda notified GM CEO Rick Wagoner it is interested in buying 6 million more shares in GM and may consider buying another 6 million shares.

In a securities filing today, Kerkorian continues to advocate for an alliance between the three auto makers.

“Tracinda continues to believe that a strong opportunity exists in a potential alliance between General Motors, Renault and Nissan, and that there should be strong General Motors board involvement in the analysis of such a potential alliance, including the utilization of independent advisors,” the filing states.

Tracinda notes that because GM owns interests in various insurance, banking trust and loan companies, the transaction, which would give Tracinda more than 10% ownership in GM, may be subject to federal and state regulatory approval.

Citing that passage, GM says it will review the filing but offers no other comment.

Kerkorian has been frustrated with the pace of change at GM. He first advocated for Wagoner to consider a tie-up with Renault and Nissan in a June 30 securities filing. GM’s board, which includes Kerkorian associate and former Chrysler Corp. Chief Financial Officer Jerome B. York, voted to begin talks with the two auto makers.

With a self-imposed mid-October deadline looming, Wagoner and Nissan-Renault CEO Carlos Ghosn had a 3-hour meeting this week to review potential synergies and cost savings. Comments from GM executives and others suggest the proposed tie-up may be in trouble.

Asked to comment on the Kerkorian development at the Paris auto show, GM Chief Financial Officer Fritz Henderson says he always welcomes investment by major shareholders. Henderson calls Kerkorian’s demands for outside review as “basically repeating what they (Tracinda) asked us to do before.”

Henderson admits there is the possibility the latest turn of events has the potential to create some hostility. But the board will have to make its own judgments on whether the review has been done fairly, he says.

The work that’s been done has been done “in the right spirit, with the right openness,” Henderson says, and he calls the board “well informed.”

The GM board meets next week and will be brought up to speed on what Henderson says has been a “thoughtful” analysis of possible synergies among the auto makers.

Henderson says Tracinda advisor Jerry York knows well his responsibilities as a GM board member and what he can and can’t do, and has behaved accordingly. He says Tracinda obviously felt it had to take its own action.