Massey was scheduled to be an official at Crewe versus Bradford in League Two on Tuesday night, but was withdrawn by the authorities in order to ensure that the controversy surrounding her did not overshadow the game.
Drury says he does not pay attention to the gender of the match officials, just their ability to do the job.
"The sex of the person in the middle doesn't affect me," he added.
"They've got big decisions to make and everyone makes mistakes.
"I don't mind whether it's a man, woman, or even an elephant refereeing a game of mine - as long as they do it properly.
"We've had Sian before and she had a fantastic game."
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