Deadline-aware concentration of synchrophasor data from spatially dispersed phasor measurement units (PMUs) is studied, with an objective to find the optimal policy for the wait time of a phasor data concentrator (PDC), such that the deadline-constrained synchrophasor data delivery from PMUs via PDC to control center achieves maximal expected throughput. When the statistical information on the communication latency from PMUs to PDC and from PDC to control center is known, the wait time problem is cast as a continuous-time optimal stopping problem. The optimal stopping policy is then shown to exist, and then obtained by utilizing the infinitesimal look-ahead rule under mild conditions. Further, the scenarios with substation PDCs or super PDCs are also considered. The dynamic wait time policy obtained by using the proposed optimal stopping approach, in comparison with two fixed wait time policies, results in significant improvement, as revealed through simulation studies.