McLeish's surprise resignation from his post as Birmingham manager on Sunday and possible appointment at Villa has sparked tension between the rival clubs. Birmingham will demand £5.4million in compensation from Villa if they try to appoint McLeish, and even plan to seek an injunction to prevent the former Scotland boss making the move.
Bevan told Sky Sports News: "If he was to go to Aston Villa or any other club I'm sure that somebody of Alex's experience and reputation will succeed, he will deliver and that's what fans are going to want, to get back up the league to where Martin O'Neill had them for three or four years."
He added: "I've spoken to Alex a number of times over recent weeks. I would expect a manager of Alex's reputation, his experience and his success in the time he's been a manager since 1994, when he's out of work to be offered a number of jobs.
"When a manager is out of work he'll look to see which job will suit him. The opportunities don't come around that often. Working in the Barclays Premier League is a goal of coaches and managers all around the world."
Bevan has called for calm, with McLeish's decision to quit his St Andrew's post sparking anger among Birmingham fans and concern among Villa supporters who fear he may not be the right man for their club, having taken Blues down from the Barclays Premier League last season.
Bevan said: "There are always, particularly with social media, going to be fans who are going to be unhappy about particular moves.
"If you go back you've got the likes of Ron Saunders who went from Villa to Birmingham and actually on to West Brom, and George Graham who went from Arsenal to Spurs. You can find many examples.
"This time when perhaps people should just calm down a bit, everyone's a little bit fraught, the reality is can the manager whoever that may be, can he deliver results, can he be given time, can he be given the right budget to achieve the goals?"
Bevan defended McLeish's right to resign by email, suggesting that was common practice and saying the LMA have "no issue with that whatsoever".
Source: PA
Source: PA