Smith's availability for the upcoming three-match series in the Caribbean, starting in Barbados on June 25, remains in major doubt.
"I'd say first Test maybe unlikely, and then go from there, but it's bit early just to tell," Australia captain Pat Cummins said after the shock World Test Championship final loss against South Africa on Saturday.
In a major blow for Australia, Smith hurt his right little finger in a crucial moment of the Lord's decider on Friday.
Wearing a helmet in the slip cordon, Smith was standing far closer to the stumps than normal when he dropped Proteas captain Temba Bavuma on just two.
Edges regularly dropped short of fielders during the Test, but this one flew towards Smith and he was unable to hang on to it.
He immediately left the field in immense pain after spilling the catch that would have put Australia in the box seat to claim back-to-back WTC titles.
Smith didn't field on Saturday as South Africa pulled off a landmark upset.
The 36-year-old left hospital to return to the team hotel on Friday night and was back at Lord's on Saturday watching on.
His finger was washed and disinfected at hospital, later placed in a splint.
The wound should heal in two weeks, and he will be able to play if he can tolerate wearing a splint while batting.
Smith will be required to keep his finger in a splint for up to eight weeks.
A compound dislocation is when bone pierces the skin, which could mean Smith needs surgery and a long rehabilitation.
One of the greatest red-ball batters of all time, Smith's possible absence will be a major blow after scoring four centuries from his past six Tests in a stunning late-career resurgence.
But Smith's misfortune could give out-of-form teammate Marnus Labuschagne a reprieve.
The 30-year-old Labuschagne averaged less than 30 in this WTC cycle over the past two years, with his last century coming back in July 2023.
A long-term No.3, Labuschagne moved up to open for the first time in his Test career for this match.
But the promotion failed to snap him out of his lean spell, making 17 and 22.
Teenage tyro Sam Konstas, who announced himself last Boxing Day, is set to be given an opportunity as opener for Australia's first tour of the Caribbean in 10 years.
If Smith can't play, Konstas could slot back in to open with Usman Khawaja, while Labuschagne would likely drop down to No.3.
But even Khawaja, who turns 39 this year, is under pressure to keep his Test career going after scores of 0 and 7 in this Test.
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