The pink to grey does not. The other one (grey to black) Fed Ex will deliver tomorrow. So I will let you know then. But I do not think that the gradient clutches have straps.
I have the talco/ardesia blow clutch (gradient beige-taupe-anthracite) and her, no strap though. But the point is, she's a clutch, right? I much prefer the grey one, the pink one is way too girly for me.
Get what fits your style!