This paper deals with the following Kirchhoff type linearly coupled system with Sobolev critical exponent
{−(1+b1‖u‖2)Δu+λ1u=u5+βv,x∈Ω,−(1+b2‖v‖2)Δv+λ2v=v5+βu,x∈Ω,u=v=0on∂Ω,
where Ω⊂R3 is an open ball, ‖⋅‖ is the standard norm of H10(Ω) and β∈R is a coupling parameter. Constants bi≥0 and λi∈(−λ1(Ω),−14λ1(Ω)),i=1,2, where λ1(Ω) is the first eigenvalue of (−Δ,H10(Ω)). Under the effects of Kirchhoff terms, we prove that the system has a positive ground state solution and a positive higher energy solution for some β>0 by using variational method. Moreover, we study the asymptotic behaviours of these solutions as β→0.