Meriam Ibrahim was sentenced in May to hang 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
children - at the airport, the sources said.
But Sudan has told the US that she and her family have now been released.
"They were temporarily detained for several hours over questions related to their documents," Marie Harf, a spokeswoman for the US state department, told journalists.
Sudan's government had assured the US that Mrs Ibrahim and her family were safe, Ms Harf added.
She said the US is working with Sudan to ensure their safe passage out of the country.
Earlier, a top Sudanese official told the BBC that although Mrs Ibrahim is Sudanese, she was using emergency South Sudanese papers with a US visa.
She would be asked to get a passport and exit visa on her release, Abdullahi Alzareg from the
ministry of foreign affairs said.
Mrs Ibrahim's husband is a Christian from what is now South Sudan and has US nationality.
One of Mrs Ibrahim's lawyers, el-Shareef Ali, told the BBC that her legal team had been denied access to her.
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