We introduce a new massive renormalization scheme, denoted mSMOM, as a modification of the existing RI/SMOM scheme. We use mSMOM for defining renormalized fermion bilinears in QCD at non-vanishing fermion mass. This scheme has properties similar to those of the mSMOM scheme, such as the use of non-exceptional symmetric momenta, while in contrast to SMOM, it defines the renormalized fields away from the chiral limit. Here we discuss some of the properties of mSMOM, and present non-perturbative arguments for deriving some renormalization constants. The results of a 1-loop calculation in dimensional regularization are briefly summarised to illustrate some properties of the scheme.