This letter proposes a novel learned feasibility estimator that considers multi-modal grasp poses for grasp and motion planning. Grasp poses inherently have multi-modal structures, that is, continuous and discrete parameters. Mixed-integer programming (MIP) is one method that solves these multi-modal problems. However, searching for all the discrete parameters costs considerable time. Therefore, by learning the feasibility of each mode from the geometric variables, the problem can be solved efficiently within a given time limit. The feasibility of grasp poses is related to the pose of the object and nearby obstacles. Utilizing this information, we introduce learned geometric feasibility (LGF), which prioritizes the integer search of MIP. LGF is scalable to multiple robots and environments because it learns the feasibility using object-oriented information. It has been demonstrated to improve the number of solved MIP problems within the time limit and to be applicable to various environmental settings.