McLeish has only one year left of his current deal and negotiations have been ongoing for several months.
Francis is aware that McLeish could be in demand after steering City to a top 10 finish in the Premier League and believes it is vital he puts pen to paper on a new deal.
The former England striker, who now works as a Sky Sports pundit, said: "The person Birmingham have got to sign up more than anyone is Alex McLeish because he has done an outstanding job.
"If you had said that Birmingham would have finished the season with 50 points, no-one would have believed that.
"I just hope there is nothing cynical in the way the contractual negotiations have taken so long.
"What I understand is one or two of the big clubs out there are looking at Alex's situation and my advice to the board is to get him signed up pretty quickly because he has had an unbelievable season.
"If it wasn't for the last six weeks, when Alex's team have slightly tailed off, he would have been in the running for the manager of the year."
In contrast, Francis believes Birmingham should not sign Ecuador striker Christian Benitez from Santos Laguna on a permanent basis.
Benitez has managed only four league goals this season and Francis feels he is not worth the fee of £6million agreed last summer if Blues want to sign him full-time.
He said: "I wouldn't spend that sort of money on Benitez.
"Earlier in the season I made a point that he creates opportunities for himself, he is an unselfish runner, creates opportunities for others with his mobility.
"But, whenever he gets in goal-scoring positions, he allows too many blocks to take place or he hits the ball against defenders.
"If you look at the last home game with Burnley, he should have come out with a hat-trick because he had so many opportunities."