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."
Source: Team Talk