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.
A top Sudanese official has told the BBC she would be freed “soon”.
Abdullah Alzareg told the BBC’s Newshour programme that Mrs Ibrahim had been arrested because she did not have the correct travel documents.
Although she is Sudanese, she was using emergency South Sudanese papers with a US visa, he said.
Her 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 is being denied access to her.
She was released from prison on Monday after an appeal court annulled the death sentence imposed on her.
She was arrested in February, and gave birth to a daughter in prison not long after being sentenced.
The family has been taken to the headquarters of one of Sudan’s security agencies, the sources said.
Before she was detained on Tuesday, Western countries had welcomed the decision to rescind the death penalty.