In his letter, Mr Ashley said he was keen to save the brand and jobs at BHS, sources have confirmed to the BBC.
The retailer is being wound down after administrators failed to find a buyer for the collapsed business.
The administrator, Duff & Phelps, confirmed it had received the letter but made no further comment.
The BBC’s Joe Lynam said he understood the firm would respond on Monday, by urging Mr Ashley to make a formal bid for the stores and/or the brand name.
In a hearing before MPs last week, Mr Ashley confirmed that he had “100%” wanted to buy the High Street retailer.
Separately, the former owner of BHS, Dominic Chappell said that just before BHS went into administration he had arranged a rescue deal backed by the billionaire owner of the sports retailer, but that it had been blocked by Sir Philip Green, from whom Mr Chappell bought BHS.
Mr Chappell said that on learning of this deal, “Philip went absolutely crazy, screaming and shouting down the phone that he didn’t want to get involved with Mike Ashley”.
However, Sir Philip has denied knowing about the deal and a spokesman has said he “was unaware of any bid interest by Mike Ashley”.