Speculation about an offer came from one of the notoriously secretive brothers, Frederick Barclay, who confirmed interest during a visit to Scotland last week.
He said the brothers also wanted to buy Mirror's 20 per cent stake in Scottish Media Group (SMG), owner of the Scottish and Grampian television franchises and The Herald newspaper.
However, on Friday, Mirror Group said its stake in the Scottish Media Group was not for sale. The papers are thought to be another matter.
The Barclay brothers' move follows the withdrawal of the German media group, Axel Springer, from a potential take-over bid and speculation that Mirror Group has restarted merger talks with regional newspaper player, Trinity.
The Mirror's Scottish titles comprise the Daily Record and the Sunday Mail. The acquisition would fill a tabloid gap in the brothers' portfolio of media interests.
Speaking to the Sunday Times, Barclay said: " "We are very interested in the Mirror [Group], especially the Scottish interests. We are working on plans, but I think that they are asking too much at the moment."
The brothers acquired The Scotsman, Scotland on Sunday and the Edinburgh Evening News in 1995 for £87m. The two have since bought both The European and Sunday Business.