The Scottish Sun claims Ralph Hasenhuttl’s side have watched the 23-year-old several times in recent weeks and are weighing up a £20 million offer.
The French under-21 international is entering the final year of his contract and Celtic are expected to cash in on their prize asset.
Leicester City are also said to be interested with former Parkhead boss Brendan Rodgers keen to be reunited with the player he helped recruit on loan and then a club-record £9 million fee from PSG.
If Edouard does move to Southampton, he will be the fifth Celtic player to have done so since 2013. The others before him, in chronological order, were Victor Wanyama (£12.5 million), Fraser Forster (£10 million), Virgil van Dijk (£13 million) and Stuart Armstrong (£7 million).