The 28-year-old midfielder, who has 13 caps, on Friday flew to England from Kaliningrad after playing in the 0-1 defeat to Belgium on Thursday, reports Xinhua news agency.
Delph had known the issue could arise when he was originally chosen for the World Cup squad, saying that his wife was "tough; she's from (the county of Yorkshire) so she can deal with it."
However, he also thanked coach Gareth Southgate for giving permission for his return to be with his wife as she gives birth.
"Gareth Southgate has been fantastic; there is nothing more important than my family for me," said Delph, who is expected to fly back after the birth to continue work as England look to book a place in the quarter-finals with a win over Colombia on Tuesday.
Meanwhile Southgate has confirmed there is no injury to central defender John Stones who was taken off at half time of England's defeat to Belgium.
The defender, who is the only outfield player to have started all of England's games, was seen with an icepack on his calf, prompting fears of a muscle injury, but the coach said there were "no concerns" and that he had always planned to replace him with Harry Maguire at the break.
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