Writing in the Daily Mail, Jordan states the Gunners should make a serious move for the former Tottenham captain.
He said: "If Arsenal fans feel deflated at the end of the season for again finishing Premier League runners-up - which they should! - I’d suggest the perfect tonic for them would be to sign Harry Kane.
"A flight of fancy? Perhaps, it depends on how deep the England captain feels his attachment to his former employees, Tottenham Hotspur.
"But professionally it would make sense for all sides. Arsenal would get the prolific goalscorer to help them take the final step and Kane would give himself the best chance of winning a league title in this country as well as surpassing Alan Shearer as the Premier League’s leading marksman.
"Both of which I’d suggest gives him a more meaningful legacy than a Bundesliga winners’ medal, which Bayern Munich achieve nearly every season.
"Certainly, if Kane decided to return to his country next season with a £54million buy-out clause there to be activated, joining Arsenal would be a more sensible decision than returning to Tottenham, who need another £300m spent on players to become realistic challengers, something that is very unlikely to happen under the current ownership.
"While there may be no economic resale value in Kane, becoming champions would open a raft of commercial benefits for Arsenal that would negate that concern."