The former Bolton Wanderers and Aston Villa midfielder, who won one cap for England, becomes Dennis Wise's sixth signing of the month following the captures of Tore Andre Flo, Armando Sa, Matt Heath, Robbie Elliott and Tresor Kandol.
Thompson made 158 SPL appearances for the Bhoys following his transfer to Parkhead from Villa in September 2000.
However, he has not played a first-team match for Gordon Strachan's side this season.