Mrs Cook told a Sunday newspaper that she would be happy to sell her shares to one of the Eye's many enemies, even Mohamed al-Fayed, owner of Harrods and of the ailing rival magazine Punch.
Peter Cook became a majority shareholder in Private Eye after the closure of his satirical club, the Establishment, in 1962.
After his death in 1995 it was thought that all his 65 per cent share would go to Lin Cook and it was reported, mainly by foes of the Eye such as the gossip columnist Nigel Dempster, that Mrs Cook would bring modern management to bear on the famously eccentric magazine.
As it turned out, Cook left 26 per cent of the magazine to his sisters, although division of the shares were subject to a legal challenge.
"I knew he [Peter] was very fond of the Eye and I was too," Mrs Cook was reported as saying. "I sort of thought Peter would not have wanted to sell the Eye. But because of their totally careless attitude towards me, I just feel that Peter loves me and my welfare would mean more to him than anything else and it would be OK."
However, it is unlikely to be: So farewell then Ian Hislop, editor/I am a banana/that was your catchphrase just yet.
The magazine's articles of association dictate that other shareholders, including Cook's sisters and Sir Dirk Bogarde and Jane Asher, be given first refusal if there is any sale.
Mr Hislop dismissed the threat of a new, hostile, Lord Gnome: "Legally she can't, as I understand that she is bound by the articles of association. And practically she can't, because I'll burn down the building."
The source of Mrs Cook's dispute with the Eye is unclear, but she has a reputation as a difficult woman and has fallen out with her late husband's relatives. She was reportedly unhappy to find a jazz band rehearsing in the magazine's offices when she first visited after his death.
Private Eye's finances are at present relatively robust, with a reported turnover of nearly pounds 3m. In contrast, Mr Fayed's reborn Punch magazine has failed to make a sales impact. Despite a recent redesign planned to give it mass-market appeal, Punch's publishing frequency was cut this month from weekly to fortnightly.