The 20-year-old has become an integral part of Steve Evans' League One side, playing in 26 of their league fixtures this season.
Tucker has been turning heads in recent weeks and is widely expected to earn a call-up to the England U21s sooner rather than later.
West Ham scouts have been raving about Tucker since they met the Gills in the FA Cup in the January. Tucker has recently signed a new deal at the Priestfield Stadium, but they could be forced to sell, should a suitable offer come in.
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