The club announced at 9pm last night they had accepted the resignations of Warburton, assistant David Weir and Head of Recruitment Frank McParland.
Ex-player Smith said his old club should move for McLeish, who won seven trophies as Rangers boss from 2001 to 2006.
“You really are in a goldfish bowl here being looked at very closely so I think you need to bring in somebody who is experienced in that respect.
“That’s why I believe somebody like Alex McLeish would be an ideal candidate.”
Rangers now face Morton in the Scottish Cup tomorrow and Smith feels some of Warburton’s summer signings have not worked out.
He added: “It’s been so confusing and it really has been a shock, especially with the timing of it, just prior to a major Cup-tie.
“They were never going to win the league this year. Celtic are stronger, they have more money than Rangers have, but I think some of the fans have been a bit annoyed about some of the performances, the signings haven’t all worked out, too.
“Some of the signings who came in, who were key signings, didn’t work out on the basis of turning the club around.
“I think that’s why it has become a bit turbulent and I think that’s why, to a certain degree, a lot of the fans will support the board in this.