Gala entered into negotiations with Basel on Monday in the hope of completing the signing of Shaqiri before the close of the the transfer window.
However, Basel were reported to have been demanding £20 million for the 20-year-old attacker, but his Turkish suitors were only prepared to spend around half of that.
"Negotiations with the professional football player Xherdan Shaqiri which began on the 23rd January 2012 have come to an end, we have not agreed terms with [Shaqiri] and his club F.C Basel," read a statement from Galatasaray.
The Switzerland international impressed against United in the Champions League this season, as Basel knocked the Barclays Premier League champions out of the competition.
Sir Alex Ferguson may now, it would seem, make an attempt to bring Shaqiri to Old Trafford, and the capture of the youngster would represent quite a coup for the United boss.