A source told the Scottish Sun: "Liam's been aware of Celtic's interest for a few weeks.
"He's unsettled at Stoke and wants to go somewhere where he will play more regularly.
"Celtic are a massive club and he knows all about them through his Republic team-mates Aidan McGeady and Darren O'Dea.
"The idea of playing for Celtic has great appeal for Liam. But the manager's situation there has yet to be established while a transfer fee and wages would also have to be agreed.
"It's up to Celtic, but there's every chance other clubs will make a move."