In this note, the relationship between linear controllability and global controllability for a class of two dimensional nonlinear control systems (x)over dot = Ax + B(x)u has been discussed in detail. It is shown that the linear controllability can imply the global controllability under some certain conditions. Furthermore, a counterexample is constructed to answer an open problem in the paper [ X. M. Hu and C. Martin, Linear Reachability Versus Global Stabilization, IEEE Transactions on Automatic Control, vol. 44, no. 6, pp. 1303-1305, 1999].