The midfielder has played under Rodgers at Swansea City and Liverpool, and has been linked with a move away from Anfield.
Celtic finally confirmed their new manager on Friday with the announcement of the appointment of former Liverpool boss Brendan Rodgers as Ronny Deila's replacement, and it would not be too much of a surprise to see a Premier League star follow him to Parkhead this summer.
Where Sam Allardyce had Kevin Nolan, Rodgers has Joe Allen, with the Welsh midfielder having played under the Northern Irishman at Swansea before being brought to Anfield by his manager.
Though Allen had already made the grade at the Liberty Stadium before Rodgers' arrival in 2010, it was under the new Bhoys boss that his career began to blossom as he played a key role in the club's first ever promotion to the top flight in Rodgers' first season in charge, and in their successful first top-flight campaign the following year.
When Rodgers left the Swans to take the Liverpool job four years ago, he convinced the Reds to pay £15 million to bring the man he has previously referred to as the jewel in his crown at Swansea to Merseyside.
Allen has failed to cement himself as a first-choice pick at Liverpool, however, and following Rodgers' sacking in October, speculation has repeatedly linked him with a move away as his contract enters its final year with no sign of an extension.
Rumours have persisted that the Welsh star could be primed for a return to Swansea, but Rodgers' appointment at Celtic could swing the pendulum heavily in Celtic's favour, if Klopp allows the player to move on in his upcoming summer revolution.
Allen could be a superb fit for the Bhoys, bringing an extra dynamism and graft to the midfield, and with promoted Rangers already turning to England to bolster their ranks, there are worse things Rodgers could try to do as one of his first acts than paying a premium to entice the star for another reunion.