Spurs meet Manchester United in Bilbao with Postecoglou's position, no matter the result, in serious doubt.
Keown told TalkSPORT: "Their performances in the Premier League, to lose 20 (21) games of football, is really quite embarrassing.
"I think that's going to cost the manager his job.
"But when he goes, he's probably going to be able to say, 'I've won a trophy in my second season, off you go.'
"He might want to put two fingers up to his own fans, to the critics."
Spurs look stronger
Indeed, Keown still rates Spurs as superior to United.
He continued: "But ultimately I see them (Spurs) winning this game.
"You could say Man United, they think their name's on it, they'll see it right through to extra-time, they'll sit as a (back) five and go to penalties and win it.
"That's the only way I see Manchester United winning it - Spurs look stronger."