We study the boundary controllability of $2 × 2$ system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be not null-controllable or null-controllable for any $T > T_0$ where $T_0 \in [0, \infty]$. Here we recover this result in the case that $T_0 \in [0, \infty)$ by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.