The Sports Direct tycoon, who recently bought the department store chain out of administration for £90 million, has been in tense talks with landlords over rents in recent weeks.
Those negotiations have resulted in 20 House of Frasers being saved from closure, safeguarding around 3,500 jobs.
However, Mr Ashley has been unable to save outlets in Edinburgh, Hull and Swindon after landlords refused to agree to new terms that would have allowed them to remain open.
Staff at these locations, which number in the hundreds, are now going through redundancy consultation.
A fourth House of Fraser in Bath is also at risk of closure as the businessman tussles with Bath City Council, the store's landlord, over new rental terms.
Mr Ashley, who also owns Newcastle United, has pledged to save around 47 of House of Fraser's 59 outlets since buying the chain.
He recently struck a deal to save House of Fraser's flagship store in Oxford Street.
Mr Ashley added: "We've showed what we can achieve on the British high street when we work together with landlords.
"I would like to thank those landlords who have helped us to rescue approximately 3,500 jobs at the stores we have saved to date.
"I am calling on everybody to pull together, including landlords and local authorities in order to help to save as many House of Fraser stores and jobs as possible on the great British high street."
Among the latest stores to be rescued is in Birkenhead, the constituency belonging to the veteran MP Frank Field.
He described the news as "tremendous".
"Six weeks ago, I wrote to Mike Ashley with an urgent plea to safeguard the future of our House of Fraser and the jobs of the store's staff.
"Thanks to the positive discussions that have taken place since then with the store's landlord, this cornerstone of our town centre is being given a chance to flourish once again," Mr Field said.
Stores earmarked for closure: Edinburgh Frasers, Hull and Swindon.
Stores that will remain open: Altrincham, Aylesbury, Birkenhead, Camberley, Carlisle, Darlington, Doncaster, Grimsby, High Wycombe, Lincoln, London (Oxford Street), Middlesbrough, Plymouth, Skipton, Telford, Huddersfield, Leeds, Maidstone, Solihull and Sutton Coldfield.