The chess federation, controlled by Mr. Ilyumzhinov, had awarded the company a no-bid contract in 2012 for the rights to organize, find sponsors and market the world chess championship, the World Cup and the Grand Prix for 11 years. The rights are potentially worth millions of dollars.

According to the memo, signed by both men, Mr. Paulson would manage the company, called Agon, and own 49 percent and Mr. Ilyumzhinov would own 51 percent. Mr. Ilyumzhinov would also provide start-up capital of up to $2 million and be repaid, with interest, out of any profits.

Mr. Ilyumzhinov and Mr. Paulson acknowledge the contract published online is real, but say it never took effect and that Mr. Ilyumzhinov never became an owner.

Mr. Ilyumzhinov, referring to the federation by its French acronym, FIDE, added in an email: “All that I sign that relates to FIDE, has to go through the board. FIDE is more than a president, it’s a huge organization. I did not and will not sign anything that is not supported and approved by FIDE.”

But this contract had not been approved by the board. The federation’s deputy president, Georgios Makropoulos, said in a statement that the deal was “just one of many proposals” and that it had been rejected. He dismissed the contract as a red herring because “it never took effect,” adding, in case anyone was wondering, “I have never received even 1 cent from Agon.”

Mr. Paulson said he wanted Mr. Ilyumzhinov to own a stake in the company to assure that the federation would try to work with Agon. “One man’s conflict of interest is another man’s alignment of interest,” he wrote in an email. “If I was going to put a lot of my money into such a speculative venture I wanted to be reassured I wasn’t going to get sheared.”