Cherries boss Eddie Howe is keen to bolster his squad for their assault on the English top flight and has reportedly earmarked Hoops utility man Mulgrew as a potential new signing.
Mulgrew was reportedly a target for Howe long before the Dean Court side won promotion to the Premier League.
The topped the Championship after beating Watford to top spot by one point and a 3-0 victory away to Charlton on the final day.
If Mulgrew moved south, he would be part of a sizeable Scottish contingent on the south coast, joining the likes of Matt Ritchie, Ryan Fraser and Josh Carmichael.
His former Parkhead team-mate Artur Boruc is currently on loan at the club from Southampton.