The former Scotland and Rangers boss believes the Parkhead stars will back interim boss John Kennedy for the role, and that could be a vital factor in Dermot Desmond’s decision-making.
Manchester United icon Keane has reportedly been in talks with the club about replacing Neil Lennon as boss.
Keane became the bookmakers favorite ahead of Sheffield United manager Chris Wilder and Manchester City youth coach Enzo Maresca.
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However, McLeish believes the big names on the Premiership side will be eager to see Kennedy given a long-term Celtic opportunity after working under Brendan Rodgers and Lennon.
McLeish told Football Insider: “If they see enough and get enough feedback from the players.
“They might say, ‘Kennedy is the man for us, Dermot.’
“To the rest of the board:” We really enjoy working with John and believe he can move us forward. ”
“You also have to have the ears of the players. They worked under Lennon, Brendan Rodgers.
“The fact that they worked under Brendan, they’ll be looking for those kinds of standards. The players will judge John Kennedy against these legendary names.