In this paper, we study the restoration of images simultaneously corrupted by blur and impulse noise via variational approach with a box constraint on the pixel values of an image. In the literature, the TV-$l^1$ variational model which contains a total variation (TV) regularization term and an $l^1$ data-fidelity term, has been proposed and developed. Several numerical methods have been studied and experimental results have shown that these methods lead to very promising results. However, these numerical methods are designed based on approximation or penalty approaches, and do not consider the box constraint. The addition of the box constraint makes the problem more difficult to handle. The main contribution of this paper is to develop numerical algorithms based on the derivation of exact total variation and the use of proximal operators. Both one-phase and two-phase methods are considered, and both TV and nonlocal TV versions are designed. The box constraint [0,1] on the pixel values of an image can be efficiently handled by the proposed algorithms. The numerical experiments demonstrate that the proposed methods are efficient in computational time and effective in restoring images with impulse noise.