Handanovic, 25, has carved a stellar reputation for himself in Serie A with Udinese and has also been targeted by Bayern Munich in the past.
The Gunners have already seen a bid for Fulham's Mark Schwarzer rejected this summer as they look to bring in a new number one and are now ready to try and lure the Slovenia international.
Portugal keeper Eduardo was also reportedly offered to Arsene Wenger, but he has now joined Genoa.
Udinese are understood to be reluctant to sell Handanovic, who is one of their prized assets, but - after excelling in Slovenia's 1-0 defeat to England at World Cup 2010 - they could be hard pushed to reject a big-money bid from a bigger club.
With question marks continuing to hover over Manuel Almunia, Handanovic could be the man to bring some stability to the club between the sticks.