The Saints are flush with cash after selling Virgil van Dijk to Liverpool for £75 million.
Dembele has had a terrific 18 months for Celtic, netting more than 40 goals for the club since joining from Fulham.
He would become the fourth Celtic player to quit the club for Southampton in recent years, following Victor Wanyama, Fraser Forster and Virgil van Dijk.
But there is a real case that Dembele might want to buck the trend and reject Southampton's advances, and 'hold out for better'.
Over the past year, Dembele has been linked with several of Europe's top clubs, with The Metro reporting Liverpool, Manchester United and Arsenal all scouted him earlier this season.
All three can offer European football, while Southampton risk getting dragged into a relegation battle after a run of poor form.
Dembele has to be wary of the wrong move, going to a club where he will sit on the bench, and Southampton are certainly a safe choice.
But after playing in Champions League clashes against the likes of Barcelona, PSG and Bayern Munich over the past year, Dembele could be forgiven for feeling a little underwhelmed in moving to a club who cannot even offer Europa League football next year.