Midlands had been in talks with the 10m-turnover group, which includes Centerprint in Nottingham, for almost a year and three weeks ago said it was poised to exchange contracts (PrintWeek, 7 June).
But it has been pipped by Atul Gandhi, who, according to former Beshara owner Hugh Tollemache, "came in and decided it was what he was looking for".
Gandhi set up AVG as a vehicle to buy Beshara.
Midlands Publishing chief executive Nicholas Humes said he felt "badly let down", and insisted he had pulled out of the bidding because he didnt feel inclined to "enter into an auction".
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