Valve held a press briefing today where they confirmed they have "three new VR-specific games in development at Valve."
Valve is developing VR hardware in tandem with these software projects. These VR-specific games are to be "new full-games" and "not smaller experiments." Details on today's briefing were shared by Gamasutra.
Interestingly, not all of their new VR initiatives are making use of their modern Source 2 Engine but reportedly also are using the Unity Game Engine for at least a portion of them.
Given Valve's VR Linux investments, these new VR games should see Linux support. More details as they become available.