This paper studies the impact of transmit antenna selection on the secrecy performance of massive MIMO wiretap channels. We consider a scenario in which a multi-Antenna transmitter selects a subset of transmit antennas with the strongest channel gains. Confidential messages are then transmitted to a multi-Antenna legitimate receiver while the channel is being overheard by a multi-Antenna eavesdropper. For this setup, we approximate the distribution of the instantaneous secrecy rate in the large-system limit. The approximation enables us to investigate the optimal number of selected antennas which maximizes the asymptotic secrecy throughput of the system. We show that increasing the number of selected antennas enhances the secrecy performance of the system up to some optimal value, and that further growth in the number of selected antennas has a destructive effect. Using the large-system approximation, we obtain the optimal number of selected antennas analytically for various scenarios. Our numerical investigations show an accurate match between simulations and the analytic results even for not so large dimensions.