Byrne enjoyed a solid debut season with Pools making over 50 appearances following his move from Halifax Town last summer.
The Irishman became one of the first names on the team sheet for both Dave Challinor and Graeme Lee and was also given the captaincy in the absence of Nicky Featherstone and Gary Liddle on occasion.
Byrne still has one-year remaining on his deal at the Suit Direct Stadium but according to reports by Football League World, the 29-year-old could now be the latest high profile player to head for the exit door.
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It comes after January signing Omar Bogle was linked with a move to Newport County recently with the striker seemingly absent from the club's warm weather training camp in Portugal.
Bogle, alongside strike partner Marcus Carver, were both unlisted in Paul Hartley’s squad to face Hibernian in their pre-season opener in the Algarve - a game which Pools suffered a 2-0 defeat in.
And although Byrne was included on Hartley’s team sheet as a substitute, he was the only player not to feature at all throughout the three 30 minutes stages of the game.
Whether the decision to keep Byrne on the sideline is linked to speculation surrounding a move to Merseyside remains unclear at this stage with Hartley preferring new signings Euan Murray and Alex Lacey as part of a back four against the Scottish Premiership side.
Rovers are on the lookout for a defender following the exits of Nathaniel Knight-Percival and Peter Clarke - whose own goal at the Suit Direct Stadium back in February helped Pools to all three points.
Micky Mellon has already added Brentford youngster Ben Hockenhull to his defensive ranks, with Byrne potentially seen as an immediate starter at Prenton Park should he sign, although Tranmere would need to negotiate a fee for the former Nottingham Forest youth defender to pry him away from the Suit Direct Stadium.
But Byrne’s exit, were it to happen, would follow a number of big names to leave Pools this summer including Timi Odusina, Gary Liddle and Luke Molyneux with Bogle’s future also uncertain at this stage.