This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap \(V(x)=x_1^2+\Lambda ^2 x_2^2\) rotating at the velocity \(\Omega >0\) , where \(\Lambda \ge 1\) and \((x_1, x_2)\in {\mathbb R}^2\) . We focus on the case where the intraspecies interaction \((-a_1,-a_2)\) and the interspecies interaction \(-{\beta }\) are both attractive, i.e, \(a_1, a_2\) and \({\beta }\) are all positive. It is shown that for any \(0<\Omega <\Omega ^*:=2\) , ground states exist if and only if \(0<a_1,\, a_2<a^*:=\Vert w\Vert ^2_2\) and \(0<{\beta }<{\beta }^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}\) , where \(w>0\) is the unique positive solution of \(-\Delta w+ w-w^3=0\) in \({\mathbb R}^2\) . By deriving the refined expansions, we further prove the nonexistence of vortices for ground states as \({\beta }\nearrow {\beta }^*\) , where \(0<\Omega <\Omega ^*\) and \(0<a_1,\, a_2<a^*\) are fixed.
