Bird strained a hamstring during the last round of the Sheffield Shield.
Sayers, 30, has yet to win a Test cap but will join the Australian squad for their four-Test series starting on March 1.
Australia captain Steve Smith says Bird’s injury ruled him out.
“It’s disappointing for Jackson,” Smith said on Tuesday.
“But it’s exciting for Chadd at the same time.
“They both stand the seam up nicely and Chadd’s been rewarded for what he has done in Shield cricket for the last couple of years.”
Smith said Bird’s injury was untimely given the Test squad would play a tour game in South Africa starting February 1.
“It’s tough news for him,” he said.
“He is a terrific guy to have around the group, he works incredibly hard.
“He’s just a great guy to have on tour with you – a terrific team man.
“I certainly feel for him that he, unfortunately, has this injury at this time.”
South Australia’s Sayers has taken the most Shield wickets in the past 18 months – he has 246 first-class wickets at an average of 24.11.
When the initial touring squad was announced last week, Sayers expressed disappointment at finding out via social media that he had been overlooked.
Sayers sought out Australia’s chairman of selectors Trevor Hohns after the squad announcement.
“I would like to know where I stand in Cricket Australia’s eyes and where I can get in their side,” he told Adelaide radio station FIVEaa.
“It’s quite hard to take at the moment. I thought South Africa would have been perfect for my bowling with the conditions over there.
“You watch Vernon Philander go about his business as he does, he takes a lot of wickets in South African conditions.”
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