The 25-year-old Republic of Ireland internationalist is in the final year of his contract and was on the cusp of moving to Hibs in August before an 11th-hour deal fell through.
The former Dundalk player’s stock is high, especially now that he is an established internationalist. However, Buddies boss Jim Goodwin is pulling out all the stops to keep him, while accepting he will have “numerous offers” elsewhere.
Goodwin said: "We're all extremely proud of what Jamie has achieved in the last 18 months.
"I watched the first hour of the Ireland game against Qatar and Jamie was excellent again. We're delighted for him.
"The kind of character he is, he could have thrown his toys out the pram when the deal wasn't allowed to happen but he's got on with things. He's shown he's willing to give his all for our jersey while he's here.
"We've made an incredible offer to Jamie, we need to make that clear. We've made him an offer we've never made a player in the history of the club.
"But we're not stupid. We know he's going to have numerous other offers and no doubt someone might come in and blow us out of the water.
"We want Jamie to know exactly what we think of him and reward him with a great contract. Whether he takes it or not, it's up to him but our supporters need to know we're not giving up hope just yet."