The static and dynamic characteristics of two-mode vertical-cavity surface-emitting lasers (VCSELs) under mode-selective feedback are investigated numerically. The model includes the spatial dependences of the optical modes, the carriers and the injection current. The research is respectively done for two different cases: weak coupling and strong coupling cases. For weak coupling case, the disc injection current is introduced and the LP01 and LP11 modes are included; while for strong coupling case, LP11 and LP21 modes are excited simultaneously by using a ring injection current. Besides, LP01 and LP11 modes are selectively reflected back by the external mirror for the weak and strong coupling cases, respectively. For short external cavities, the laser keeps its continuous wave (CW) operation. The modal power varies periodically with respect to the external cavity length and increases (or decreases) with the increase of the external mirror reflectivity. The variation trend of modal power is determined by the external cavity length, which decides the relative phase of the reflected light and the in-cavity one. Moreover, we find that the two modes under consideration exhibit anti-phase behaviors with the variation of the feedback conditions. Especially, for the strong coupling case, the enhancement of one mode can result in the thorough suppression of the other mode. For long external cavities, however, the relaxation oscillation process of laser becomes undamped, which induces rich nonlinear dynamics for both the reflected mode and the one without feedback. Through computer simulation, many typical dynamics, such as the CW, period-one, period-two, period-four, quasi-period, and chaotic states, are observed for both modes. It is also shown that fixing the feedback level, the dynamics shown by the laser exhibit periodic evolution with respect to the external cavity length; however, for a given external cavity length, a distinct period-doubling route to chaos is found with increasing external mirror reflectivity. Therefore, although just one mode is directly affected by the external feedback, the other mode still exhibits the similar nonlinear behaviors due to the carriers dynamics induced by the mode with feedback.