The 29-year-old was an integral member of the Adler XI en route to the Europa League crown last term but now only has one year left to run on his contract.
Tottenham Hotspur and West Ham have both been linked with a move for Kostic, and the Hammers are said to have already seen a bid worth £15m including add-ons turned down.
According to the Daily Mail, West Ham will soon return with a higher bid, as David Moyes is particularly keen to welcome Kostic to the London Stadium.
Kostic - who faced West Ham in the Europa League - is represented by the same agent as Gianluca Scamacca, whose move to the Hammers was confirmed earlier this week.
Over in Italy, Juventus could attempt to sign another midfielder in the summer transfer window after new recruit Paul Pogba suffered a serious knee injury in training.
The 29-year-old will spent at least a few weeks on the sidelines recovering, but he could be out for as long as four to five months depending on which rehabilitation approach he chooses.
According to El Larguero, via Calciomercato, Atletico Madrid's Saul Niguez has now emerged as an option for the Old Lady as Diego Simeone seeks to offload the Spaniard.
The report claims that Saul could arrive at the Allianz Stadium on a season-long loan move, and Roma have also been offered the chance to sign the Chelsea flop.
The 27-year-old endured a nightmare stint at Stamford Bridge under Thomas Tuchel last season, contributing just one goal in 23 outings in all tournaments.
Pogba has already completed a move to his former club this summer, and Paris Saint-Germain's Idrissa Gueye is now being tipped to do the same.
The 32-year-old was a regular under Mauricio Pochettino last season but could struggle for the same opportunities in Christophe Galtier's 3-4-1-2 formation.
Vitinha, Marco Verratti, Ander Herrera and Danilo Pereira are all competing for places in the midfield, and the Ligue 1 champions are still pursuing Lille's Renato Sanches.
Gueye is also out of contract in 2023, and journalist Fabrizio Romano claims that Everton have touched base to enquire about re-signing their former midfielder.
The Senegal international spent three years on the books at Goodison Park between 2016 and 2019, scoring four goals and setting up five more in 107 games for the Toffees.
Finally, Ademola Lookman's search for a new club could supposedly lead him to Serie A, with RB Leipzig keen to cut their losses on the 24-year-old.
Lookman has been sent out on loan to both Fulham and Leicester City since signing for Die Roten Bullen from Everton in 2019, and he scored six Premier League goals for the Foxes last term.
A separate update from transfer expert Romano adds that Atalanta BC are now on the verge of sealing a deal for Lookman, and talks are in the final stages.
The attacker has allegedly already agreed personal terms with Gian Piero Gasperini's side, who will secure his signature after talks with Arsenal over a deal for Nuno Tavares collapsed.
Lookman has only made 24 appearances for Leipzig over the past three years, racking up five goals and four assists in the process.