We present new observations with the Karl G. Jansky Very Large Array of seven X-ray-selected tidal disruption events (TDEs). The radio observations were carried out between 9 and 22 years after the initial X-ray discovery, and thus probe the late-time formation of relativistic jets and jet interactions with the interstellar medium in these systems. We detect a compact radio source in the nucleus of the galaxy IC 3599 and a compact radio source that is a possible counterpart to RX J1420.4+5334. We find no radio counterparts for five other sources with flux density upper limits between 51 and 200 μJy (3σ). If the detections truly represent late radio emission associated with a TDE, then our results suggest that a fraction, >åisebox-0.5ex 10%, of X-ray-detected TDEs are accompanied by relativistic jets. We explore several models for producing late radio emission, including interaction of the jet with gas in the circumnuclear environment (blast wave model), and emission from the core of the jet itself. Upper limits on the radio flux density from archival observations suggest that the jet formation may have been delayed for years after the TDE, possibly triggered by the accretion rate dropping below a critical threshold of i̊sebox-0.5ex 10$^-2$-10̂-3 dotM_Edd. The non-detections are also consistent with this scenario; deeper radio observations can determine whether relativistic jets are present in these systems. The emission from RX J1420.4+5334 is also consistent with the predictions of the blast wave model; however, the radio emission from IC 3599 is substantially underluminous, and its spectral slope is too flat, relative to the blast wave model expectations. Future radio monitoring of IC 3599 and RX J1420.4+5334 will help to better constrain the nature of the jets in these systems.