Inter were stuck for over a year discussing a deal with goalkeeper Handanovic, only for him to finally pen fresh terms in January 2016.
But his contract is already nearing its conclusion, with the agreement set to end in 2019.
Inter are desperate to keep their goalkeeper but the draw of the Premier League almost saw him move to England two summers back.
And reports in Italy claim Inter are determined to avoid the same scenario.
According to FCInterNews, Inter rejected interest from both Liverpool and Tottenham in the summer.
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Liverpool manager Jurgen Klopp eventually stuck with goalkeeping duo Loris Karius and Simon Mignolet, although question marks hang over the futures of both players.
Spurs were concerned Hugo Loris could be subject to interest from abroad, with Real Madrid and Paris Saint-Germain seeking new shot-stoppers.
FCInterNews claim the player’s agent Fali Ramadani met with Piero Ausilio to discuss the possibility to Handanovic leaving for Liverpool or Spurs before deciding not to act on the interest.
They now want the 33-year-old Slovenia international to extend his contract by a further year so as to avoid further transfer speculation over the coming months.