Henry, who is Arsenal’s record goalscorer, is currently managing MLS side CF Montreal after being appointed their head coach in November 2019.
Bournemouth, however, are now keen on possibly hiring the Frenchman and they are seeking permission from Montreal to speak to Henry.
The Cherries have spoken to a number of candidates since sacking Jason Tindall earlier this month, including Henry’s Arsenal team-mate Patrick Vieira.
Vieira, unlike Henry, is currently out of work after being sacked by Ligue 1 side Nice in December.
Tindall was sacked with Bournemouth sat sixth in the Championship and former Tottenham defender Jonathan Woodgate has been in interim charge.
The club are looking to hire a permanent successor and they want to speak to Henry about the position.
Before working at Montreal, Henry previously took charge of Monaco and he also enjoyed a stint working as part of Roberto Martinez’s backroom staff at Belgium.