Spurs acted swiftly to replace Pochettino, sacking their manager after five years at the helm before appointing Jose Mourinho.
While Mourinho's previous spells in charge of Chelsea look likely to make him a divisive figure among the Tottenham faithful during the early stages of his reign, Pochettino opting to switch allegiance in north London would cause a whole other kind of ructions.
Gunners manager Unai Emery has come under increasing scrutiny amid a mixed start to the season and, although Keown does not want any rash decisions on the former Sevilla and Paris Saint-Germain tactician's future, he feels the Arsenal board should be duly aware of unexpectedly changing circumstances.
"There's no need to panic at Arsenal, we are getting close to panic by the way, because we're not having the best of seasons," he told the Daily Mail. "But the powers that be at the top of the club should be seriously considering Pochettino.
"I have a lot of admiration for what he did at Tottenham, the way he nurtured those young players and he didn't spend a great deal of money."
"I would have stayed with Pochettino. I think he deserved that. I think it's a hugely significant sacking. You're going to have managers now looking over their shoulders.
"I know the poor record he has recently, but I still would have given him another chance. I don't like the way Tottenham have done it.
"There will be a lot of chairmen looking at him now. Manchester United wanted him this time last year. I think Arsenal would have to be interested if they want to take a step forward."
Mourinho cut a contented figure on his first day as Spurs manager and Keown believes the embattled figure from his final days in charge of Manchester United must be consigned to the past.
"He lost the dressing room and fell out of love with the game of football in his quest to be successful," the former England international said. "He has to jettison that part of himself."