Former Livingston owner Angelo Massone claims he has lodged a bid for Hearts on behalf of an Italian consortium.
He says the formal offer - reported to be £4.5million - has been submitted to Hearts owner Vladimir Romanov and a response is awaited.
Hearts were issued with a winding-up order last week over a tax bill of almost £450,000, although the club and HMRC have agreed a deal to extend the deadline for payment.
Massone told the Scottish Daily Mail: "Together with the other investors, we have written to Mr Romanov's lawyer with a formal offer.
"We do not know how much Mr Romanov wants for the club. But I believe there is no one else in Scotland who can match the value of the offer we have made.
"I am willing and able to bring serious, wealthy, respected investors to Scottish football. We want to save Hearts and develop the club and make them a strong force once more. But we also want the supporters to have democratic representation on the board and be the real owners of the club."
The Foundation of Hearts and Supporters Direct Scotland have had discussions with the hierarchy at Tynecastle over possible future fan ownership.
Massone added: "If we are not successful with our bid, then I am willing to help Hearts supporters to own the club instead. If Foundation of Hearts or Supporters Direct need financial help in terms of money to realise the dream of the fans owning the club, then I will speak seriously with them."