In this paper we study the algebra L([sum ]) generated by links in the manifold [sum ]×[0, 1] where [sum ] is an oriented surface. This algebra has a filtration and the associated graded algebra LGr([sum ]) is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra ch ([sum ]) of chord diagrams on [sum ] to LGr([sum ]).

We show that multiplication in L([sum ]) provides a geometric way to define a deformation quantization of the algebra of chord diagrams on [sum ], provided there is a universal Vassiliev invariant for links in [sum ]×[0, 1]. If [sum ] is compact with free fundamental group we construct a universal Vassiliev invariant. The quantization descends to a quantization of the moduli space of flat connections on [sum ] and it is natural with respect to group homomorphisms.