The talks are being led by Andrew Neil, editor in chief of the Barclays' newspaper group, European Press Holdings, which also includes The Scotsman, Scotland on Sunday, The Midlothian Enquirer and Sunday Business.
Under the proposed rescue operation for The European, a second party would take an equity stake in the company. The Barclay twins would maintain a stake and Mr Neil would become the third shareholder.
Bloomberg, the US-based financial information company, has been cited as one of the three companies currently negotiating with Mr Neil, but the company declined to comment.
The European, founded in 1990 by Robert Maxwell, has swallowed up an estimated pounds 50m since the Barclays bought it in 1992. Mr Neil, the former editor of The Sunday Times who became editor in chief of The European in May 1997, changed the publication to a tabloid and plans to turn it into a business magazine.
These have, however, brought him into conflict with the company's chief executive, Bert Hardy. The paper's circulation has fallen from an initial print run of 340,000 to an official 132,000, although the real figure is probably less than 100,000, according to insiders.
Mr Neil's idea is to establish a joint venture with a company that has the know-how to remarket and distribute the title - in magazine form - across Europe, an idea to which Mr Hardy is thought to be bitterly opposed.
Such a joint venture would be a shrewd face-saving move for the Barclays, who would otherwise find themselves having to suffer either the humiliation of closing the paper or selling it wholesale to someone who could equally embarrass them by making it a success, say sources close to the negotiations.
Critics, however, have accused Mr Neil of contributing to the paper's decline because of his staunchly anti-European Union stance. An article in the French newspaper Le Monde this week described Mr Neil as a "frenetic Francophobe" and the current editor of The European, the former Tory MP, Gerry Malone, as a "raving Europhobe".