Boro hope the Republic of Ireland defender will agree a loan deal with a view to a permanent move.
St Ledger stayed behind at the final whistle to applaud the Preston fans following Saturday's 2-0 victory over Swansea.
Alan Irvine: "We know that there is an interest from Middlesbrough and we know that their interest is quite strong.
"Unfortunately nowadays the decision (to keep a player) doesn't always come down to the manager.
"Every player at this club will have his price, regardless of whether I want him to stay or whether the chairman and the board want him to stay.
"Sean St Ledger has been fantastic for us and the last thing I want to do is lose a good player.
"He's come on in leaps and bounds and is a lad who is desperate to learn and desperate to improve.
"He's very self-critical and is always looking to correct anything he does wrong.
"We've got a fantastic relationship and I hope that continues for a long time."