The player’s current contract runs only until January meaning he could be available for less than someone of his talent would command.
Southampton is a popular destination for Celtic players with Stuart Armstrong, Victor Wanyama, Fraser Forster and Virgil van Dijk all having moved to the south coast in recent years.
According to the Scottish Sun, the Saints will watch Christie at the Euros for Scotland.
The 26-year-old provided one of the most memorable moments of the season with an emotional interview after Steve Clarke's men had defeated Serbia on penalties to reach the tournament.
For Celtic he hit just eight goals in 48 appearances.
Back in October Christie said: “I've still got plenty of time on my current contract.
"Before this one I was down to five months to go at Celtic until we signed a new deal, so there's plenty of time for that to resolve itself.”