The Sun says Arsene Wenger has already held talks with Cole over filling the gap left in the Gunners midfield.
Chelsea star Cole met Gunners boss Wenger before jetting out to South Africa with England last week.
The former West Ham ace, 28, is out of contract and set to quit Stamford Bridge after seven years.
Assistant boss Ray Wilkins admitted: "Carlo and myself would love Joe to stay. But it's down now to the club to see what we can do."