The Cardinals will fly to Florida following Monday's game to open a three-game series with the Marlins on Tuesday night.
Thurston, who had been awaiting word from his wife on their child's pending arrival for the past week, would have had to make two-cross country trips had he gotten the call while the team was in Florida.
But instead of waiting it out, Thurston decided to head back to California, and doctors are now scheduled to induce labor, likely on Tuesday.
"He will probably miss Tuesday and Wednesday," said manager Tony La Russa. "But we should have him back for Thursday."
With a noon ET start time on Thursday in Florida, La Russa said Thurston was scheduled to take a red eye flight on Wednesday night to ensure that he is back in time.
Tyler Greene got the start at third base on Monday, but La Russa said that had nothing to do with Thurston's pending departure. Greene, Brian Barden and Brendan Ryan are all candidates to play third in Thurston's absence.
"That's more than enough," La Russa said.
Thurston entered Monday hitting .248 with one home run and 16 RBIs but was in a 1-for-14 skid at the plate, and he went 0-for-2 on Monday.
B.J. Rains is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.