After leaving the house, the cabinet went into an emergency meeting.
The move could lead to the third dissolution of parliament in the oil-rich Gulf state in three years.
If Sheikh Sabah al-Ahmad al-Sabah, the emir of Kuwait, accepts the resignations, he can form a new cabinet or dissolve parliament establish new elections.
Sheikh Sabah is yet to make an official statement on the resignations.
Three Islamist MPs called for the questioning of Sheikh Nasser, a nephew of Sheikh Sabah, last week.
Iranian leader's entry
They accused Sheikh Nasser of allowing a prominent Iranian Shia leader to enter Kuwait despite a legal ban.
They have also accused him of failing to "perform his constitutional duties ... and that it was time that Kuwait had a premier capable of running the state and achieving the wishes of the people".
Additionally, the trio said that corruption and squandering of public funds has increased under Sheikh Nasser's leadership.
Sheikh Nasser is a prominent member of the ruling family. Impeaching such individuals is considered unacceptable.
It is possible that Sheikh Sabah could now suspend parliament and the constitution, meaning no fresh elections will be called.
The Al-Sabah family has controlled Kuwait for its complete history, spanning back about 250 years. Few local people question the family's rule.
Saad al-Anezi, Al Jazeera's correspondent in Kuwait City, said: "The emir of Kuwait is the only one who has the power to constitutionally make a decision [now].
"The scene is very, very chaotic here in the parliament. Everybody, even the speaker doesn't know what is going to happen - we just spoke to him a short while ago.
"Everybody is extremely concerned that this time the dissolution of the parliament will be unconstitutional, which means that it will be dissolved and elections will not be called within two months, which is a constitutional constraint.
"Everybody is worried it might lead to confrontation between the opposition and the government."