The Championship outfit have made contact with Henry's current club CF Montreal for permission to speak with the Arsenal legend about filling their managerial vacancy.
The Cherries have been without a boss since axing Jason Tindall at the start of this month.
First-team coach Jonathan Woodgate has been in interim charge.
The ex-England defender has won three of his four games and will manage the side for Saturday's clash with QPR.
But Henry, 43, has emerged as the front-runner to succeed Tindall.