According to Buddha's characteristics and teaching in Pali Canon, could he have taught Vajrayana (and Karmamudra)?

One of the characteristics of Vajrayana is its secret transmission (esoteric tranmission) of certain teachings. The other one is the controversial karmamudra (sexual practice to reach spiritual attainment). So the question is, according to the characteristics and teaching of Buddha in Pali Canon, could he have taught Vajrayana and karmamudra?