The Saints are hoping to pip a number of their Premier League rivals to the 21-year-old's signature, with interest in his services having mounted steadily.
They have, however, seen the highly-rated midfielder snub their first contract proposal.
Wanyama has admitted that he turned down the initial proposal from St Mary's and revealed that he is prepared to listen to offers from other sides now that Celtic have opened the exit door.
"While the bid was okay for Celtic, the terms were not right for myself and my agent," he said in the Southern Daily Echo.
"I am open to what offers are made, but I am relaxed and it's important that Celtic are happy too."
Cardiff City are reportedly ready to jump to the front of the transfer queue as they step up their search for new recruits, while Stoke City and Liverpool continue to watch events unfold.