The pair crossed paths when covering England's win over San Marino for ITV last week and Rangers hero McCoist raised the question of a possible move to Glasgow.
Keane is among the favourites for the Celtic Park vacancy – with suggestions former Manchester United team-mate Nicky Butt is being lined up as his number two.
But McCoist revealed that the 49-year-old Irishman was giving nothing away.
Speaking on talkSPORT, McCoist said: "He effectively ignored me really.
"Roy just kind of looked at me as if to say 'yeah, and I'm really going to tell you the answer'.
"I got that, you know the stare, the world-famous stare – I was on the receiving end of it at Wembley on Thursday night!
"And I still don't know whether he's going to get the Celtic job or not. Effectively, that's where we are on it."