A UAE official has reportedly said work permits for the teachers were valid only for Dubai and were invalid in Abu Dhabi.
Iran said the teachers in Iranian schools had been regularly working in Abu Dhabi in recent years using their Dubai credentials.
A court hearing for the nine teachers scheduled for Monday was postponed, Iran's ISNA news agency said.
The teachers, all male, were arrested on Wednesday in the city of Al Ain, according to Fars news agency.
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But Rasoul Papayi, an education ministry public relations official, denied the report about female teachers, ISNA reported.
The Emirati charge d'affaires was summoned on Sunday night in the absence of the ambassador.
The foreign ministry protested against the arrest of the nine male teachers and demanded their immediate release, state media said.
"The arrest of Iranian teachers in the Emirates and their release is being followed up and consultations are under way," said Khalilollah Babalou, head of the international department of Iran's education ministry.
There are 10 such schools in the UAE, which has a large Iranian population, estimated in the hundreds of thousands and concentrated in Dubai, a regional trading hub.