Tierney, 22, has been the subject of strong interest from Arsenal and Napoli this summer, a topic which is showing no signs of dying down.
Fresh reports have claimed The Gunners are preparing a second bid of £22million, however Celtic are reportedly holding out for at least £3m more.
But while The Hoops retain a firm stance over Tierney, McLeish says the full-back is ready for a new challenge at a 'more important club'.
“Tierney is ready for the big jump and to go to a more important club than Celtic," the 60-year-old told CalcioNapoli24.
"When his team played Barcelona, Manchester City or Paris Saint-Germain in the Champions League, he always stood out, and that makes him ready.
"He’s got attacking qualities, but is also a great defender."
The Italian radio station McLeish was speaking to confirmed there was genuine interest from Napoli in Tierney but admits no deal is close.