The 33-year-old midfielder is in talks with the Govan club over a three-year contract and visited Ibrox and the club’s Murray Park training ground on Saturday.
Barton was a key player in Burnley’s Sky Bet Championship triumph but was reportedly only offered a one-year extension at Turf Moor and wants a longer deal.
McLeish, speaking at a media event at Hampden Park, organised by William Hill, sponsors of the Scottish Cup, told Press Association Sport that Barton still has plenty to offer for a club who return to the Ladbrokes Premiership next season after a four-year absence.
“It is a really interesting one,” he said. “I heard it from a taxi driver last night and I said, ‘Och, I’ve heard all these stories before, it will be a lookalike or something’.
“But that would be a very ambitious signing and I think he would bring a lot to Rangers.
“I know Joey well, I did some television work with him in the Middle East and he knows the game really well.
“He is a fiery character and he will probably do it in a tactical way now.
“I am sure there will still be a red card or two in there somewhere but I am also pretty positive that it would be a kind of controlled aggression that he would produce.
“He is a fit chap. He is 33 but I don’t believe his legs have gone, he has just helped Burnley win the league and that is the only thing you worry about when players get to their 30s.
“Some players’ legs go before others but I don’t believe that is the case with Joey Barton.
“He would bring vision, calmness and certainly drive to the dressing room.”
McLeish, back in the UK after a spell as boss of Egyptian side Zamalek, “can only see one winner” in next week’s William Hill Scottish Cup final between Rangers and Hibernian at Hampden Park and that is Mark Warburton’s men.
Alan Stubbs’ side are trying to recover from their Ladbrokes Premiership semi-final play-off defeat to Falkirk on Friday night where they went out to a last-gasp goal from Bob McHugh which sent the Bairns into the final on a 5-4 aggregate.
McLeish believes the Hibs boss would do well to spend the week brainwashing his players into believing they can bring the trophy back to Leith for the first time since 1902, an achievement he would rate alongside Leicester City winning the Barclays Premier League this season.
The former Hibernian, Aston Villa and Scotland manager, who lost the final with the Hibees in 2001 and won it with Rangers in 2002 and 2003, said: “As the week unfolds Alan Stubbs will be working on his players’ confidence and belief and has got to try to get it out of their head by Tuesday or Wednesday and really focus on producing an incredible result.
“It would be of Leicester City proportions if he could do it.
“My approach to it would be to give Hibs every bit of brainwashing that I could, to highlight the evidence of the games that they have done really well against Rangers throughout the season.
“That is what he must focus on.
“Alan will have his own ideas - he might have that idea as well - but certainly he will do it his own way and it makes for a pretty intriguing final.”