The 28-year-old, who is the vice-captain of a new Australian side after a series of retirements, broke the thumb on his left hand in the one-day match against England at Lord's on Sept. 5 and X-rays last week confirmed the bone was still broken.
"I saw the surgeon at the four-week mark, which was last Friday," Warner told reporters in Sydney on Tuesday. "I had an X-ray and it was still broken.
"He said generally around the six-week mark, I'm coming up to five weeks at the moment, generally it would be healed.
"I'll have a follow up X-ray this Friday and then if all goes well I'll be having a hit (in the nets) on Monday."
Warner was hoping to be fit enough for New South Wales' first class match against South Australia from Oct. 28-31, a week before the first test against Brendon McCullum's side starts in Brisbane on Nov. 5.
Australia's test side will be named during the round of matches that Warner is hoping to play in and the batsman said he felt if he makes that Sheffield Shield game, he should be fine to play the test the following week.
"My goal is to play that Shield game, I want to play that Shield game to have a hit because I don't think I'll be able to get out here and play any of the last couple of games for the (domestic one-day competition)," Warner added.
"You don't lose that (ability) overnight.
"You're only one or two hits away in the nets, an hour off getting back to what you can do. But obviously batting against bowlers is going to be the key.
"I've broken this before and I know exactly what I have to do. That's getting it right with the bat first and then we'll worry about the fielding and catching later."
(Reporting by Greg Stutchbury in Wellington; Editing by Patrick Johnston)
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