We consider the confluent hypergeometric function M(a, b; z) for z is an element of C and Rb > Ra > 0, and the confluent hypergeometric function U(a, b; z) for b is an element of C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b; z); one of them in terms of incomplete gamma functions gamma(a,z) and another one in terms of rational functions of e(z) and z. We also derive a convergent expansion of U(a, b; z) in terms of incomplete gamma functions gamma(a,z) and Gamma(a,z). The expansions of M(a, b; z) hold uniformly in either Rz >= 0 or Rz <= 0; the expansion of U(a,b;z) holds uniformly in Rz > 0. The accuracy of the approximations is illustrated by means of some numerical experiments.