The 35-year-old damaged his right elbow while throwing in the field for the Sydney Sixers in their washed-out Big Bash League derby against the Sydney Thunder.
Smith will seek further advice from a specialist, with the intention of flying to Dubai to join the Test team in preparing to face Sri Lanka.
He is not Australia's only injury concern, with left-arm spinner Matt Kuhnemann set to miss his first Test opportunity in almost two years after surgery on his fractured right thumb.
Kuhnemann, who played three Tests in India in 2023, was injured playing for the Brisbane Heat in the BBL last Thursday.
He will stay in Australia until his surgical wounds heal, but is planning to start bowling again in the coming days.
Working in his favour is the surgery happened on his non-bowling hand, giving him an outside chance of playing in Sri Lanka.
The first Test begins in Galle on January 29, with the second match from February 6.
Captain Pat Cummins, who is missing the Tests due to the birth of his second child as Smith skippers Australia in his place, continues to recover from soreness in his left ankle.
A Cricket Australia spokesman said Cummins would "likely require a period of physical preparation and recovery".
Cummins was picked in Australia's preliminary squad for the Champions Trophy in Pakistan, with chairman of selectors George Bailey confident the star quick would be fit and available for the 50-over tournament.
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