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The Scottish Football Association is on the look-out for a new manager for the national side after sacking Craig Levein on Monday.
Bookmakers have installed Gordon Strachan as the favourite to succeed Levein, but there have been calls for Jordan to be considered for the position.
Jordan enjoyed a glittering playing career with AC Milan, Leeds and Manchester United, but he has not managed a team on a permanent basis since 1997 when he was in charge at Bristol City.
Jordan has spent most of his recent years as Harry Redknapp's number two at Portsmouth and Tottenham, but McLeish thinks the 60-year-old would have no problem making the step up to international management.
"Joe Jordan is a fantastic candidate," McLeish told BBC Radio Five Live's Sportsweek programme.
"He is tactically tremendous because of working with Harry all these years and with the different systems they have played.
"I don't think Joe would be caught short in terms of tactical nous at that level."
McLeish, who also thinks former Celtic manager Strachan would do a good job if appointed, managed Scotland for 10 months with relative success before quitting to join Birmingham, who were then in the Barclays Premier League.
There was an initial clamour for the 53-year-old to take the job again after Levein's dismissal, but he insists he would prefer to take charge of a top-flight English side instead after his recent disappointing spells in charge of Birmingham and their neighbours Aston Villa.
"It's not that I wouldn't like the job. If I was offered the Scotland job... it's a very difficult job to turn down, but I feel I have unfinished business in England and I like the day-to-day stuff," the former Rangers manager said.
McLeish believes Levein, who won just three of his 12 games in charge of the national side, should perhaps have got more out of what he regards as a decent team.
"I believe they should have done better," McLeish said.
"I do think the current crop are a decent bunch. James Morrison started in the Premier League for West Brom and scored a great goal yesterday.
"Some of the guys haven't reached their level yet at international level.
"They haven't produced that extra couple of per cent that exists in every player on the planet."