Warburton left Rangers in freakish circumstances on Friday evening, with the club releasing a statement saying they had accepted the resignations of him, assistant David Weir and head of recruitment Frank McParland but Sky sources understand the three "categorically deny" quitting.
McLeish, who was manager for five years until 2006, insisted there has been no contact with his former club.
McLeish was at Ibrox on Sunday as part of the Sky Sports' team that covered the home side's 2-1 win over Morton, setting up a Scottish Cup quarter-final against either Dunfermline or Hamilton.
Warburton, assistant David Weir and head of recruitment Frank McParland all left their posts on Friday night in farcical circumstances. It is purely coincidence that I'm here today.
The former Brentford manager and his associates are set to take the Ladbrokes Premiership club to court seeking compensation believed to be in the region of £1 million and will be represented by the English League Managers Association.
"You can say that to our players, we have to have those attributes and with our quality, I think we can go all the way".
"I need to wait and see if Rangers want to talk to me first".
"Obviously, there is a lot of speculation, but at the moment there is no update".
McLeish won two Scottish titles as Rangers manager and the Glasgow club also won four knock-out trophies during his tenure - the Scottish Cup twice and League Cup three times.
"My focus is really on doing the job that I am employed to do at St Johnstone".
"That is my focus, to maximise what I have got at St Johnstone".
Rowett might be seen as a similar candidate to Mark Warburton, having limited managerial experience but achieving relative success in his short time to date.