Amla, who last week retained his place at the top of the individual test batting rankings, played in last week's first test in Abu Dhabi but flew home on Friday for the impending birth.
Team officials had hoped the baby would arrive over the weekend to allow Amla, who scored a century in the first innings, to rejoin his team mates in the United Arab Emirates for the second test, which starts on Wednesday.
"Hashim's wife hasn't given birth yet, so he won't be playing in tomorrow's test," South African team manager Mohamed Moosajee said.
Steyn faces a late test of his fitness after a hamstring scare in the nets on Monday.
"Dale sustained what was initially thought to be a right hamstring strain but fortunately the MRI scan revealed that there was no major pathology so we are treating it as tightness," Moosajee added in a statement.
"He will undergo a fitness test. The challenge with a tight hamstring is the fear of aggravating it. It could easily become a strain or progress into a tear, so we will be managing it carefully."
South Africa are likely to replace Amla with left-hander Dean Elgar and have Rory Kleinveldt on standby should Steyn fail the test. Pakistan lead the two-match series after a seven-wicket win last week.
Pakistan's first Test victory ended a run of 15 matches without a loss for the Proteas, who had not played in the Test arena for 10 months.
"It is important to find a solution," said vice-captain AB de Villiers.
"We are the number one ranked team in the world and we are going to have to go out there and play like that.
"Technical changes can't happen, we have to believe and have a fresh mindset that we are going to level the series."
South Africa failed to pass 250 in either innings at the Sheikh Zayed Stadium and new coach Russell Domingo called it "a soft performance from the batting unit".
Pakistan coach Dav Whatmore said he was not surprised by the win despite an upset loss to Zimbabwe last month.
"I knew that we could be competitive in different conditions and with a slight change of personnel. We gave a good competitive effort over the four days," Whatmore said.