In　this　paper,　we　consider　the　continuous-time　nonzero-sum　constrained　stochastic　games　with　the　discounted　cost　criteria.　The　state　space　is　denumerable　and　the　action　space　of　each　player　is　a　general　Polish　space,　while　the　transition　rates　and　cost　functions　are　allowed　to　be　unbounded　from　below　and　from　above.　The　strategies　for　each　player　may　be　history-dependent　and　randomized.　Models　with　these　features　seemingly　have　not　been　handled　in　the　previous　literature.　By　constructing　a　sequence　of　continuous-time　finite-state　game　models　to　approximate　the　original　denumerable-state　game　model,　we　prove　the　existence　of　constrained　Nash　equilibria　for　the　constrained　games　with　denumerable　states.