Body centered cubic (bcc) Ti-based alloys are of interest for multiple technological applications ranging from aerospace technology to biomedicine. However, these alloys are usually unstable at low temperatures. Indeed, the calculated elastic modulus C′ of bcc Ti–V alloys with low V concentrations is negative at 0 K temperature, indicating their mechanical instability. Here, we investigate elastic moduli of the Ti–V system in the vicinity of mechanical instability theoretically and experimentally. Our calculations predict that mechanical stabilization of bcc Ti–V alloys, which is governed by the hardening of C′, is possible at as low V concentration as 18 at. %. We synthesize single-phase bcc alloys with as little as 22 at. % of V with low values of Young’s modulus. Moreover, we predict strong concentration dependence of anisotropy of Young’s modulus in these alloys that can also be used in tuning the alloy composition to design materials for specific applications.