Tunneling magnetoresistance was found to be suppressed with decreasing temperature for magnetic tunnel junctions (MTJs) oxidized under high plasma power. A strong temperature dependence of the junction resistance was observed, along with zero-bias anomalies of dynamic resistance at low temperatures. Resistance shows a logarithmic dependence on temperature, and resistance versus temperature exhibits a scaling behavior. Our experimental data can be explained in a consistent way by the Kondo effect in the MTJs with the Kondo temperature TK=20-30 K.