The 38-year-old player-coach has been offered a contract to stay at Glanford Park, but has also attracted the itnerest of Third Division rivals Darlington.
Now Linfield have thrown their name into the hat for his signature.
Scout Jim Emery told the BBC: "He would be a big asset for Linfield but Scunthorpe are keen to keep him."
The former Man City and Everton star has scored 44 times in his three seasons in North Lincolnshire.
Weekend reports linked Beagrie with the assistant manager's position at Scottish Premier League outfit Hibernians.