Sport says there's discussions at board level over the prospect of replacing Valverde this summer - with Henry emerging as the favoured option.
Henry gets along with Barca president Josep Maria Bartomeu and is good friends with former France teammate Eric Abidal, the club's sporting director.
For now, there's no immediate plans to remove Valverde. The discussions haven't gone beyond the boardroom in order not to unsettle the coach and his players.
Meanwhile, Barca will have to cough up if they're to move for Henry as he only signed with the Impact a month ago for the coming MLS season.