The Hammers are considering both loan and permanent moves for the 30-year-old Serbia international, who is valued by Juventus at around £25million.
West Ham tried to sign Kostic last summer before he moved to Turin from Eintracht Frankfurt.
Kostic, who is yet to feature in Serie A this season, has three years left on his contract at Juventus and West Ham are due to continue talks on Thursday.
The Hammers also want to sign a striker before Friday’s transfer deadline and are considering alternatives targets after a move for Corinthians forward Yuri Alberto collapsed.
West Ham director of football Tim Steidten has been in Brazil this week but a deal could not be agreed.
Other options being considered by West Ham include Paris Saint-Germain striker Hugo Ekitike.
West Ham have also looked at Peterborough defender Ronnie Edwards, though a move is considered unlikely.
Everton are interested in a late move for Maxwel Cornet, who could be allowed to leave if West Ham sign Kostic or another midfielder.