We present two results for bounding marginally outer trapped surfaces (MOTSs) in Killing initial data satisfying the null energy condition and containing an untrapped barrier. The first one applies to the stationary case and states that no bounding MOTS lying in the exterior region where the stationary Killing vector is causal and penetrating into the timelike region can exist. The second result applies to the static case and shows that no bounding MOTS can penetrate into the exterior region where the static Killing vector is timelike. These results extend an interesting theorem by P. Miao (Miao 2005).