The 28-year-old has been heavily linked with a move away from San Siro this summer, with the Premier League - not La Liga - cited as his most likely destination.
Gunners boss Arsene Wenger is on the lookout for a new goalkeeper to provide competition for current number one Wojciech Szczesny, and has been credited with an interest in Handanovic.
However, Arsenal's hopes of bringing the Slovenia international to the Emirates Stadium have been dealt a blow after Barcelona emerged as admirers.
Tito Vilanova is eyeing a long-term replacement for Victor Valdes who will leave Camp Nou when his contract expires in 2014, and has identified Handanovic as a possible successor.
Inter Milan are demanding €30million - or £25.5million - for the former Udinese stopper, though, so Barcelona must raise their offer substantially if they are serious about signing him.
News of Barca's bid might also spark Arsenal into action, forcing the north London club's had in order not to miss out on another of Wenger's summer targets before the 2013/14 campaign starts in August.