In image restoration and reconstruction applications, unconstrained Krylov subspace methods represent an attractive approach for computing approximate solutions. They are fast, but unfortunately they do not produce approximate solutions preserving nonnegativity. As a consequence the error of the computed approximate solution can be large. Enforcing a nonnegativity constraint can produce much more accurate approximate solutions, but can also be computationally expensive. This paper considers a nonnegatively constrained minimization algorithm which represents a variant of an algorithm proposed by Kaufman. Numerical experiments show that the algorithm can be more accurate and computationally competitive with unconstrained Krylov subspace methods.