Aston Villa have rejected bids of £12million and £15million from the Reds for Downing and McLeish wants the England international to remain at the midlands club.
McLeish says he overcame failing to secure a move to Manchester United as a player and would expect the same response from Downing if he is still at Villa next season.
He said: "I've spoken to Stewart, I've told him I don't want to sell him.
"There's an obvious distraction there for him but he's been a credit to himself and applied himself in the right way and worked his socks off in training. I don't see a problem professionally.
"Sir Alex Ferguson wanted to sign me for Manchester United when I was at Aberdeen and that wasn't possible in the end.
"That was a big disappointment but I got my head down and worked and I don't doubt Stewart has the character to get through this is he stays at the club.
"It's part of your growing up as a footballer."
Villa will demand £19-20million if it becomes apparent they cannot retain Downing, who has two years remaining of his current contract.
McLeish is optimistic of signing a new keeper to replace Brad Friedel in the near future with Shay Given expected to complete a £3.5 million move from Manchester City.
He said: "We certainly need a keeper and we are making progress. We would like it to happen a bit quicker but we are moving forward."