The 29-year-old initially arrived at Die Adler on loan from Hamburg in the summer of 2018 and has proven instrumental on the left-hand side for the club since.
Kostic has racked up 33 goals and 63 assists in 170 outings for Frankfurt in all tournaments over the past four years, including seven goals and 15 assists from 43 games this season.
Three of Kostic's goals and six of his assists came in Frankfurt's run to the Europa League crown, and he assisted Rafael Borre's equaliser in the final against Rangers.
However, Kostic's contract at the Deutsche Bank Park is due to expire in the summer of 2023, and it remains to be seen if he will be offered an extension by the Europa League champions.
Should the Serbia international decide against penning fresh terms, Frankfurt could be pressured into a sale before watching him walk away for nothing next year.
According to The Express, Tottenham scouts were in attendance to watch Kostic in action during the Europa League final, and the Lilywhites are now readying a £25m offer.
The report adds that Spurs initially took a liking to the former Groningen and Stuttgart man back in January, but Antonio Conte's side decided against making a move.
Despite the uncertainty surrounding his long-term future, Frankfurt manager Oliver Glasner does not believe that Kostic has played his last game for Die Adler.
"He still has a year's contract. We spoke on the field, but I won't tell you what [we said]. I think that was not his last game [for Eintracht Frankfurt]," Glasner told reporters in Seville.
Kostic's exploits in the wing-back area are believed to have caught the eye of Conte, who has struggled to fill that role on a consistent basis throughout the season.
A new left-sided defender is therefore thought to be on the Spurs' transfer wishlist, with Brighton & Hove Albion's Marc Cucurella and Inter Milan's Ivan Perisic also linked with moves to North London.
Tottenham put five unanswered goals past Norwich City on Sunday to seal a return to the Champions League on the final day of the Premier League season, forcing Arsenal to settle for fifth place.