Boehly is part-owner of Major League Baseball side Los Angeles Dodgers, and is keen to take over a Premier League club.
According to the Telegraph, Chelsea is his top target.
But senior club sources continue to insist that the Blues are not for sale, and that current owner Roman Abramovich won’t enter talks.
Boehly is keen to make it work, having already been part of a group that tried to buy Tottenham Hotspur in 2014.
His initial approach was rejected, but the American will make another attempt when he believes the time is right.
Boehly’s enquiry was the latest of at least four that Abrahmovic has turned down over the last year-and-a-half.
The Russian has stood firm despite being unable to work in the UK because of visa issues.
His valuation of the club is said to have been inflated by the recent investment in a 10 per cent stake of Manchester City, which valued the Premier League champions at £3.74 billion.
Boehly, who is worth around £5 billion, was asked by Bloomberg in September how confident he was about buying a club.
“You can never predict these things,” he said.
“One of the nice things about football clubs is that they’re relationship-driven and team driven, and therefore it’s unlikely one just gets put up and sold by auction.
“So you develop long-term relationships and get to know them, but ultimately whether or not something happens I don’t know. But what I do know is I’ve learned a lot.
“I’ve learned a lot about the fan-base, the global opportunity. It’s the best product in the world.”
He added: “The billion-pound question is the price. It’s hard to buy quality and not have to pay up.
“Then it’s a question of can you continue to build on what you’ve acquired at that price? That’s the question.
“I continue to believe there is a global opportunity for the best clubs. Ultimately, it comes down to can something get done that both sides think makes sense?”
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