The Scottish champions have been linked with the 22-year-old numerous times over the past two seasons.
Their chance to finally land the goalkeeper could come this summer as Newcastle United look set to move on from the player who came through the club’s academy.
Woodman has made just one appearance for the Magpies and is set to enter the final 12 months of his contract.
However, Arsenal, Everton and Fulham are also said to be interested in snapping up the England under-21 international.
Woodman has previously impressed in the Scottish Premiership, first with Kilmarnock in 2016-17 and then with Aberdeen last season.