The Wallabies head to HQ this weekend looking for a measure of revenge after being whitewashed by the Red Rose during their summer tour Down Under.
Cheika and opposite number Eddie Jones got into a war of words during England's time in Australia and have now renewed hostilities in the build-up to the final autumn international of the year.
Jones recently stoked the fires by requesting a meeting with Saturday's match referee Jaco Peyper to discuss what he feels is a questionable scrummaging technique by the Australians.
Cheika has now hit back by claiming Jones has damaged his reputation in his home country with his 'vitriolic' comments about the Wallabies, while also suggesting his former Randwick team-mate should take a look at his own pack, in particular Cole, before throwing around accusations.
"He wants to talk about our scrum because we're cheating in the scrum - an important thing to note is he's got to look at his own players," Cheika said.
"They are the ones who have a prop with a yellow card. That same prop's been infringing the law since his career started probably, if not all of this year.
"If he thinks the refs are that naive that if he has a go at our scrum they'll forget about his guy, that's up to the ref. It's not up to me. Nothing I can say will make a difference about that. It's up to the referee whether he gets influenced by that.
"The guy [Cole] has been boring in and falling down all of June, that's his tactic. If he thinks that me saying something back on that is going to change the referee's mind, that's totally up to the ref."