The Hoops’ Teesside-born boss Tony Mowbray also wants to sign Preston defender Sean St Ledger, who was on loan at Boro for three months.
But Preston have described Celtic’s £1.75m offer as “derisory”.
Boro were ready to pay up to £4.5m for Irish international St Ledger, though this included the initial £1.5m they agreed to pay as part of the loan arrangement.
St Ledger allegedly had a bust-up with Strachan and the Irish international insisted he would not sign for the Teessiders.