Australia lost the three-match series when defeat in Melbourne on Saturday gave England a 2-0 lead with a game to play and Hansen said Cheika's refusal to engage with Jones through the media had ended up playing out on the pitch.
"Cheika's not come back, he's letting Eddie have a free rein -- to the point where it actually seems like he's letting Eddie bully him in the media," Hansen, whose team defeated Australia in the World Cup final last year, told New Zealand media.
"I don't know if that's because they know each other that well, or if there's a pecking order from the old days. That's gone on to the park, hasn't it?"
Cheika allowed Jones to hog the headlines in the lead-up to each of the first two tests but was not about to let New Zealander Hansen's jibe slide.
"I'd be pretty nonplussed about that," Cheika told reporters in Sydney on Tuesday when asked about the remark.
"That's a really shallow sort of view I suppose on what's really involved in coaching and he knows better than that.
"It's easy to kick blokes when they are down, so we are down, but we'll be getting back up, don't worry about that.
"I think that's a place where sometimes a lot of people like to see us, especially in rugby.
"Australia is down, we've been down before and we've got back up and we will be doing the same again right now."
Jones has relished ramping up the traditional sporting rivalry between England and former colony Australia and said his side's win in Melbourne was "rope-a-dope" against a blunt Wallabies attack.
The Australian suggested Hansen's comment had ulterior motives.
"He's only being mischievous, he's trying to help himself for the Rugby Championship," Jones told reporters in Sydney, referring to the southern hemisphere tournament involving Australia, New Zealand, South Africa and Argentina.
Although having taken a pot-shot at Australia's set-piece and not missed an opportunity to describe Cheika as the 'world's best coach' at every media appearance, Jones denied trying to get under his rival's skin.
He again showed his penchant for a sound-bite, however, referring to Saturday's dead rubber at Sydney Football Stadium as like a "World Cup final" and declared England were the side to "put a challenge to (the All Blacks)".
"To win a World cup you've got to win three big games in a row," he said.
"You've got the quarter-final, semi-final and final, so it's a great dress rehearsal for us."
(Reporting by Ian Ransom; Editing by John O'Brien)
