The 27-year-old's Muslim family have said they'll take her to court to try and stop her leaving the country.
She was freed from death row in June after being charged with marrying a Christian because Islamic law believed she was a Muslim.
Meriam claims she was raised Christian by her mother.
Last week her father dropped a legal bid to formally establish her as his Muslim daughter.
Police arrested her as she tried to leave the country claiming she was using false travel documents.
She's staying at the US embassy with her husband and two children.
The Reuters news agency says the new case will be heard by a Sudanese judge on 4 August.