Warner, the team's vice-captain, hurt his neck while taking a high catch at training this week, but Smith said his condition improved significantly overnight.
"He's very confident and he says he'll be right to go," Smith told reporters ahead of Thursday's Ashes opener at the Gabba.
"He'll be OK. It's part and parcel of playing cricket. Guys have injuries every now and then and have little niggles.
"He's improved a fair bit over the last 24 hours and hopefully he can keep improving and be 100 percent at match time."
Smith said selectors had been discussing a replacement option to be on standby, but he was confident the player would not be required.
"There's a good chance someone will be in as cover but I'm not sure who at this stage... they're thinking about it at the moment," he said.
Fast bowlers Chadd Sayers and Jackson Bird were left out of the team after being chosen in the original 13-man squad.
Australia: David Warner, Cameron Bancroft, Usman Khawaja, Steve Smith (capt), Peter Handscomb, Shaun Marsh, Tim Paine, Mitchell Starc, Pat Cummins, Josh Hazlewood, Nathan Lyon.
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