Moeen, 31, made the accusation in his soon-to-be-released autobiography, where he also labelled the 2015 Australian touring party as "rude" and "disrespectful".
"It was a great first Ashes Test in terms of my personal performance. However there was one incident which had distracted me," Moeen wrote.
"An Australian player had turned to me on the field and said, 'Take that, Osama'. I could not believe what I had heard. I remember going really red. I have never been so angry on a cricket field," he wrote.
"I told a couple of the guys what the player had said to me and I think (England coach) Trevor Bayliss must have raised it with Darren Lehmann, the Australians' coach. Lehmann asked the player, 'Did you call Moeen, Osama?' He denied it, saying, 'No, I said, 'Take that, you part-timer'," Moeen wrote.
"I must say I was amused when I heard that, obviously I had to take the player's word for it, though for the rest of the match I was angry," he further wrote.
"We have a clear set of values and behaviours that comes with representing our country.
"We take this matter very seriously, and are following up with the ECB (England and Wales Cricket Board) as a matter of urgency to seek further clarification around the alleged incident," he added.
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