In [19] Scheinberg generalized a celebrated theorem of Carleman by showing that a continous function onRncan be approximated with arbitrary speed by entire ffunctions ofncomplex variables. Alexander showed in [1] that an unbounded smooth curve inCnis such a set of Carleman approximation. Frih and Gauthier in [8] studied Carleman approximation by entire functions inCnon products of curves inCn, n1+…+n>k=n. In the present paper, we show that a product set in a product of Riemann surfaces is a Carleman set if and only of each projection is a Carleman set, thus generalizing the results of [18] and [19].