Desmond was among the ever-growing field of bidders looking to buy the Telegraph Group, which includes the Daily Mail & General Trust, Axel Springer, Gannett, the Barclay brothers and Collins Stewart.
It is understood that Desmond's holding company Northern & Shell, which owns Express Newspapers, was asked to increase its bid to stay in the race but refused to go any higher.
It had already been speculated that Desmond might drop out because of investment bank Lazard's attempts to seek bids for the whole of Hollinger International. Desmond had already said that like the Barclay brothers he was not interested in bidding for Hollinger's other assets, the Chicago Sun-Times and the Jerusalem Post.
With Northern & Shell out of the race it now leaves open the question of West Ferry Printers, which Express Newspapers owns jointly with the Telegraph.
A spokesman for Northern & Shell said: "We were in the bidding process to buy a business not a trophy asset. We look forward to talking to the Telegraph's eventual owners about the printing arrangements at West Ferry."
Express Newspapers has the right to buy out the Telegraph Group if there is a change of ownership, which seems likely. The Telegraph Group's contract to print its newspapers at the plant expires in 2009 and Desmond has suggested this will not be renewed if a rival bid succeeds.
During his pursuit of the Telegraph Group Desmond sold off his adult magazine business, Fantasy Publications, for around £20m to Remnant Media in a move that was widely interpreted as an effort to boost his respectability in the battle for ownership of the Telegraph.
As well as the various media groups bidding a number of venture capital firms including the Blackstone Group, Bain Capital, Thomas H Lee, Apollo, Spectrum, Providence Equity, Saban Capital, Apax Partners, 3i Group PLC and Candover Investments are also all thought to be considering bids.
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