|Meriam Ibrahim gave birth to a baby daughter in prison|
Meriam Ibrahim was sentenced to hang in May for renouncing Islam, sparking widespread outrage at home and abroad.
About 40 security agents detained Mrs Ibrahim - along with her husband Daniel Wani and two sons - at the airport, the sources said.
On Monday, Mr Wani said the family intended to leave for the US.
Mrs Ibrahim was released from prison after an appeal court annulled the death sentence imposed on her.
She was arrested in February, and gave birth to a boy in prison not long after being sentenced.
Born to a Muslim father, Mrs Ibrahim, 27, married Mr Wani, a Christian, in 2011.
Sudan has a majority Muslim population, and Islamic law has been in force there since the 1980s.
Even though Mrs Ibrahim was brought up as an Orthodox Christian, the authorities considered her to be a Muslim.