In this paper we introduce the class of [∞,C]-isometric operators and study various properties of this class. In particular, we show that if T is an [∞,C]-isometric operator and Q is a quasi-nilpotent operator, then T + Q is an [∞,C]-isometric operator under suitable conditions. Also, we show that the class of [∞,C]-isometric operators is norm closed. Finally, we examine properties of products of [∞,C]-isometric operators.