Marcus Harris, Cameron Bancroft and Matt Renshaw are seemingly locked in a three-way selection battle ahead of a two-Test series against the West Indies in January.
There have also been calls to move one of Travis Head, Marnus Labuschagne or Mitch Marsh to the top of the order, which would open the door for allrounder Cameron Green to return to the line-up.
Warner on Tuesday publicly nominated Harris as his preferred choice to take over, prompting a stinging response from Cox, who was on Australia's Test selection panel from 2006 to 2011.
Well bugger me - another modern first! I have never heard of a current player anointing their successor. What's wrong with 'that's a question for George Bailey not me'? I need a lie down…! 🤷�♂� — Jamie Cox (@jamiecox1969) https://t.co/p1QGxZfP2TDecember 26, 2023
"Well bugger me - another modern first!" Cox posted on social media platform X, formerly known as Twitter.
"I have never heard of a current player anointing their successor.
"What's wrong with 'that's a question for George Bailey not me'? I need a lie down …!"
Cox's comment came as he reposted part of the quote from Warner's response to a journalist's question about who should replace him at the top of the order in January.
It is not clear whether Cox saw or heard Warner's full response, in which the 37-year-old deferred to selectors before offering his personal opinion.
"Look, it's a tough one. It's obviously up to the selectors," Warner said on Tuesday night, after the close of play on day one in the second Test against Pakistan.
"But from my position I feel like the person who's worked their backside off and has been there for a while in the background (is Harris).
"Harry's been that person, he's toured ... he scored a hundred the other day and he missed out in a couple of other games, but he's always been that person who was next in line.
"If the selectors show their faith in him then I'm sure that he'll come out and play the way he does.
"He's not too dissimilar to me. If he sees it in his areas he goes for it and plays his shots, and I think he would fit well."
Australian captain Pat Cummins sidestepped the controversy when asked about a possible replacement for Warner after the close of play at the MCG on Wednesday.
"I am not a selector. I obviously speak to them, but we'll work that out after the Sydney Test," Cummins said.
"My honest answer is I haven't really been part of any conversation. I think everyone just knows we've got a bit of time."
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