It is believed that the Kenyan international has agreed to terms to a reported fee of £12million after the two sides held talks on Saturday.
Celtic who bought the 21-year old for just £1m from Belgian side Beerschot will make a huge profit and also will secure the biggest transfer fee to be paid to a club in the Scottish Premier League.
Wanyama's move to Southampton will mean that they have secured their second major signing of the Summer with Lyon centre back Dejan Lovren joining earlier in the week.
In other Southampton transfer news, Roma's Pablo Osvaldo has turned down the chance to join the Saints after they made a reported offer of around £17m to buy the out-of-favour forward who linked up with current manager Mauricio Pochettino during their days at Espanyol.