Romero is Tottenham’s top defensive target and the club have held talks with Atalanta about taking him to north London.
However, Spurs have a number of other centre-backs on their radar and they are admirers of Fiorentina’s Milenkovic, who is in the final year of his contract.
Milenkovic has made over 120 appearances for the Serie A side and he is expected to be sold this summer if he does not sign a new deal.
However, if Tottenham sign Romero then they will pull out of the chase for Milenkovic, according to Italian daily La Nazione.
Spurs are working to push a deal for Romero over the line and success will signal the end of their efforts for Milenkovic.
Tottenham are also keen on Bologna defender Takehiro Tomiyasu, but any transfer is only expected to take place following the Olympic Games.
Spurs boss Nuno Espirito Santos is looking to build a solid backline for his maiden season in charge, especially in the wake of the departure of Toby Alderweireld.