The confining quark matter (CQM) model, in which the confinement and asymptotic freedom are modeled via the Richardson potential for quark-quark vector interaction and the chiral symmetry restoration at high density is described by the density-dependent quark mass, is extended to include isospin dependence of the quark mass. Within this extended isospin-dependent confining quark matter (ICQM) model, we study the properties of strange quark matter and quark stars. We find that including isospin dependence of the quark mass can significantly influence the quark matter symmetry energy, the stability of strange quark matter and the mass-radius relation of quark stars. In particular, we demonstrate that although the recently discovered large mass pulsars PSR J1614.2230 and PSR J0348+0432 with masses around two times solar mass (2M⊙) cannot be quark stars within the original CQM model, they can be well described by quark stars in the ICQM model if the isospin dependence of the quark mass is strong enough so that the quark matter symmetry energy is about four times that of a free quark gas. We also discuss the effects of the density dependence of quark mass on the properties of quark stars. Our results indicate that the heavy quark stars with mass around 2M⊙ (if exist) can put strong constraints on isospin and density dependence of the quark mass as well as the quark matter symmetry energy.