The Premier League side, who were promoted in the summer, are hoping to bring in the 23-year-old.
O’Riley, who netted 19 goals and provided 18 assists, helped Celtic win the Scottish Premiership last term.
The Scottish Sun are reporting that the bid from Southampton did not come close to Celtic’s valuation.
While the Hoops are willing to sell, they will only do so on their terms, per manager Brendan Rodgers.
He said: “Every player has a value, but we see him as a really important player, and while he’s here, we’ll continue to develop him and improve.
“Celtic is a club where you can move on but you don’t have to, and it’s a very difficult club to move on from.
“But naturally, with the wages that are on offer for players, then that makes it a real challenge for Celtic to compete at that level.”