The England midfielder has been linked with a move to The Emirates after saying he will not sign a new contract at Stamford Bridge when his current deal expires at the end of June.
He has also been a target for Manchester United, Tottenham, Manchester City and West Ham.
But Chelsea assistant coach Ray Wilkins insists he and manager Carlo Ancelotti will be trying to persuade Cole to stay.
Wilkins, speaking on BBC Radio 5 Live's Sportsweek programme said: "We've obviously been in dialogue with Joe for months now and it's pretty evident that Carlo and myself would love Joe to stay but it's down now to the club to see what we can do.
"We sincerely hope Joe stays with us - he's a smashing guy and he's a wonderful footballer so we'll keep our fingers crossed."