"Cricket South Africa (CSA) will, in all likelihood, grant Graeme a leave of absence," South Africa team manager Mohammed Moosajee said.
Smith will return home after the first Test that starts on July 23 at the Oval to be with his wife Morgan Dean and go back before the second Test which begins in Leeds on August 2.
Moosajee said he did not believe that Smith's short absence would have any adverse impact on the team, since he would return in time to lead the team out for the second Test.
"The style of leadership (that coach) Gary (Kirsten) is trying to create is to divide the responsibility on as many shoulders as possible," Moosajee told the daily SportsDay.
Commenting on how fatherhood might change Smith's performance, Moosajee said, "I think he's looking forward to being a father and I think he'll make a damn good one because he's got the right value systems."