Let us clarify the terminology, keeping in mind that we are discussing particle physics which is modeled quantum mechanically and obeys special relativity rules.
Particles are modeled by their four vectors, and the "length" of the four vector is the invariant mass characteristic of the particle. What is conserved are the energy and momentum between the incoming particles and the outgoing ones in an interaction.
Virtual particles exist only within the Feynman diagram representation of the interaction of elementary particles, based on quantum field theory. In the above diagram the particle labeled "virtual" is an internal line with a variable four vector as constrained by the limits in the integral which the diagram summarizes. It is the incoming and outgoing momentum and energy that are conserved. The virtual "particle" is off mass shell, and it is called a "particle" because it carries the quantum numbers of the name, but not the characteristic mass.
( If you want to see how a crossection is calculated this way see here, example 6)
So it has no meaning to ask about an instantaneous value of the energy momentum of the virtual particle, as it scans the available phase space under the integral.
BTW the relativistic mass is not used when discussing particle physics seriously. It is only useful for thought experiments with space ships :). In particle physics, where virtual exchanges exist in the models, one uses four vectors and their invariant masses.