"There's no need for panic at Arsenal, we are getting close but the people at the top of the club should be considering Pochettino", Keown said while speaking at a Sportsmail event in London.
The Gunners are now sixth in the table - eight points off fourth-placed Manchester City - and Keown says the club's board needs to consider head coach Unai Emery's future.
Whether Pochettino would even consider moving to his old club's rivals however remains to be seen.
Keown's belief that Arsenal should snap up Pochettino following his Tottenham departure is shared by Arsenal fan - and Spurs-goader - Piers Morgan.
"Unai Emery's side start the day eight points off fourth place in the table and I think this is a must win game for the Gunners' boss given Saints sit just one point and one place off the foot of the table".
Jose Mourinho arrived as the new head coach less than 12 hours later, but Keown says the Argentine deserved more time.
Talking on talkSPORT Redknapp, a former Spurs boss, said, "With Pochettino going from Tottenham, there will be one or two Premier League managers waking up this morning thinking, "oh my god, this ain't too clever". Real Madrid would take him, I think Arsenal would have to look at him".
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Of course, I am willing to help any appropriate law enforcement agency with their investigations, if necessary. The prince said he had sought and acquired his mother's permission to stage again from royal duties.
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