The midfielder flew to England yesterday to talk over personal terms with the Barclays Premier League club who finished 14th last season under new manager Mauricio Pochettino.
If a deal can be struck then Celtic are in line to make a £10m profit on the Kenyan international who they signed from the Belgian club Beerschot two years ago for just £900,000.
Beerschot are understood to be due a percentage of any fee accrued by Celtic as part of a sell-on clause written into his contract.
The fee agreed between Celtic and Southampton would be the highest ever involving a Scottish club, beating the £12m Rangers paid for Tore Andre Flo in 2000. The previous highest transfer fee paid to a Scottish club was the £9.5m Spartak Moscow shelled out to sign Aiden McGeady from Celtic in 2010.
Wanyama, 21, had an outstanding season with Celtic, helping Neil Lennon’s side to a league and Scottish Cup double and a place in the last 16 of the Champions League.
Lennon will now hope some of the money comes his way to further strengthen his squad, although he has already spent around £4m on Portuguese striker Amido Balde and Dutch defender Virgil van Dijk.
The Celtic manager has been credited with an interest in Wolves striker Kevin Doyle and free agent James McFadden.