The motion, which accuses Ali Asghar Fani of being incapable of performing his duties, faces a vote in the conservative-dominated parliament on June 24, the official IRNA news agency reported.
As in past impeachment actions, the effort to dismiss Fani may lead Rouhani to parliament to argue in his minister's defence. A "yes" vote requires a simple majority of the chamber's 290 MPs.
In early March, several hundred teachers demonstrated in Tehran and other cities to demand increased wages. Fani, a reformist, was approved by parliament in October 2013 after Rouhani's first choice for the post was rejected.
Rouhani, who is pushing for a nuclear deal with world powers that would lift the sanctions imposed on Iran in recent years, also faced a veiled attack from a senior regime figure on Tuesday.
Mohsen Rezaie, an Iran-Iraq war veteran and current secretary of the Expediency Council, an advisory body to supreme leader Ayatollah Ali Khamenei, recently rejoined the country's elite Islamic Revolutionary Guards Corps which he once headed.
"Let them not say we have no clean drinking water, that the country is encircled, that we cannot stand on our feet," said Rezaie, adding "that the message of the martyrs" was that Iran will never surrender.
But the president received backing from one of his predecessors.
Akbar Hashemi Rafsanjani, a senior establishment figure, in a speech at a public event said Rouhani "faces a minority who can grandstand and which can exert power throughout the country".
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