The 24-year-old has enjoyed a fruitful season for the Hoops, scoring 11 goals and claiming eight assists from 51 appearances.
Despite becoming a focal part of Brendan Rodgers's all-conquering side, McGregor is rumoured to be pining for a move south of the border to try his luck in the Premier League.
According to The Sun, the Scotland international is being courted by the Cherries, who are reportedly bracing themselves to lose several key men in midfield including Harry Arter.
Watford, who have former Bhoys striker Tommy Johnson on their books as a scout, is another potential destination for McGregor, who has previously played in England with Notts County.
McGregor has three years left on his Parkhead contract and is rumoured to have a £10m price tag.