The billionaire said in a stock market update that an offer to buy Debenhams' Danish business Magasin du Nord for £100million, revealed last week, is one of several offers that Sports Direct has made to put the department store chain on a more secure footing.
Sports Direct added today that its proposals are better than the "multiple insolvency processes" Debenhams is considering as it looks to restructure.
"The offer for Magasin is one of several offers that Sports Direct has made to provide the board of Debenhams with a valid alternative to its apparent view that multiple insolvency processes are required to address Debenhams' current liquidity concerns and to facilitate a wider balance sheet restructuring," the sportswear firm claimed.
As part of the plan, Newcastle United owner Mr Ashley, who owns just under 30% of Debenhams, would become chief executive of the ailing high street firm.
For its part, Debenhams has put in motion plans to secure £200million in new funding from its lenders and pursue a restructuring plan which would wipe out existing shareholders, including the Sports Direct founder.
Sports Direct said this is "not a workable solution".
But Debenhams has dismissed Mr Ashley's alternative proposals, arguing that they do not address the company's funding and restructuring requirements, and adding that the tycoon's appointment as chief executive would be a conflict of interest given his ownership of House of Fraser.
Responding to those claims, Sports Direct said it does not consider House of Fraser to be a competitor of Debenhams.
"In any event, were Mr Ashley to become CEO of Debenhams, he would, as previously announced, step down from his roles at Sports Direct. He would also be subject to fiduciary duties to Debenhams," it said.
"Sports Direct would strongly recommend that the Debenhams board reconsiders the offers made by Sports Direct to date and their own duties as directors of Debenhams."
Mr Ashley also lashed out at a "lack of engagement" and the "limited access to information" provided to him by Debenhams.