The 31-year-old is on the fringes of Thiago Motta's plans in Turin at the start of the 2024-25 term, failing to make a competitive appearance under the new boss at the Allianz Stadium.
Kostic was a regular fixture for the Old Lady during the 2023-24 campaign, playing 29 matches in Serie A, with 26 of those coming from the start of games.
A 64-time international since making his senior debut in January 2015, the veteran represented Serbia at this summer's Euro 2024 in Germany.
Since securing a move from then-Europa League holders Eintracht Frankfurt to Juve during the summer of 2022, Kostic has turned out in 87 matches for the Turin-based club, scoring on three occasions.
Southampton 'join' race for Juventus star Kostic
According to Tutto Juve, three clubs are battling for the services of Juventus' Kostic before the closing of the summer transfer window on Friday night.
The report claims that Premier League newcomers Southampton are one of the sides interested in securing the signature of the 31-year-old over the next 24 hours.
As well as the Saints, it is understood that there is additional interest from an English club in the form of Oliver Glasner's Crystal Palace.
There is also a lucrative route into the Saudi Pro League on the cards for Kostic, who has admirers at state-backed Al-Ahli in the Middle East.
Juventus are supposedly open to selling the Serbian before the window slams shut, with the Italians said to want at least €7m (£5.8m) for the winger.
Where would Kostic fit in at Southampton?
During their promotion-winning campaign of 2023-24, Russell Martin employed a 4-3-3 formation with Southampton, who looked to control their second-tier contests with the ball.
Now facing the challenge of the Premier League, the Saints have changed to a 3-5-2 setup, aiming to deal with the attacking firepower of top-flight opponents.
Kostic would be able to provide a left-footed option at left wing-back, a role which Kyle Walker-Peters has assumed in the defeats to Newcastle United and Nottingham Forest.
The 31-year-old has a wealth of experience of competing at the top level of European football, knowhow that would be like gold dust in a side littered with Premier League newbies.