In this study, the frequency of the external cavity diode laser (ECDL) is locked to the hyperfine transition of ⁸⁷Rb D₂ line 5²S_1/2, F=2→5²P_3/2, F=3 using modulation transfer spectroscopy (MTS) frequency stabilization. The laser linewidth is narrowed from 382.18 kHz in the free-running mode to 37.94 kHz after frequency stabilization. The narrow linewidth laser after frequency stabilization can be utilized as the probe light for the integrating sphere cold atom clock. Thus, the contribution of the laser frequency noise to the short-term instability of the atomic clock could be smaller than 5.6×10^-14 τ^-1/2.