The Sun says the French striker has been targeted to launch the newly-formed Seattle team when they make their MLS debut next year.
And the franchise's multi-billionaire owner Paul Allen will tempt Henry with a deal worth £150,000 a week.
But the Spanish giants are planning a summer clear-out after slumping to third place in La Liga.
And Henry could go despite having more than three years left on his £128,000-a-week deal.
Henry has 14 goals in 38 games but is playing on the left wing.