This paper addresses the hedging problem of American Contingents Claims (ACCs) in the framework of continuous-time Itô models for financial market. The special feature of this paper is that in the financial market the investor has to face fixed and proportional transaction costs when trading multiple risky assets. By using the auxiliary martingale approach and extending the results of Cvitanic and Karatzas [Cvitanic J, Karatzas I. Hedging and portfolio optimization under transaction costs: a martingale approach. Math Finance 1996;6:135-65] on pricing European contingent with transaction costs in the single-stock market, an arbitrage-free interval [hlow, hup] is identified, and the end points are characterized by auxiliary martingales and stopping times in terms of auxiliary stochastic control problems. Here hup and hlow are so-called the upper hedging price and the lower hedging price.