Reports suggest that the Midlands side turned down a £15 million bid from Liverpool for the England international earlier this week.
Speculation is rife that Liverpool will return with a £20 million bid but former Birmingham City manager McLeish is loathe to losing another key member of the squad after waving goodbye to Ashley Young and Brad Friedel earlier in the transfer window.
McLeish has also revealed he has spoken to Downing regarding his future and plans to do so again once pre-season training starts again.
“He had a magnificent season last year and really came to full maturity as a top player,” McLeish told the Birmingham Mail.
“I understand why there's teams interested in him and that is what happens when you play well consistently.
“His performances last season were top drawer. That's why he's not for sale. The club don't want to sell him.
“I spoke to him briefly last week and I intend speaking to him again when all the lads are back in for training.”
Twenty-six-year-old Downing, who has two-years left to run on his Villa deal, scored eight goals in all competitions last season.
By Ben McAleer
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
