Netanyahu's government is pushing ahead for a parliamentary vote this week on a centrepiece of the overhaul - a law that would give the governing coalition the final say over all judicial appointments.
Tens of thousands of protesters have taken to the streets, military and business leaders have spoken out against it and leading allies of Israel have voiced concerns.
Defence Minister Yoav Gallant, a senior member of Netanyahu's Likud party, became the first to break ranks late on Saturday by calling for the legislation to be frozen. Gallant cited the turmoil in the ranks of the military over the plan.
He was sacked on Sunday.
Earlier on Sunday an Israeli good governance group asked the country's Supreme Court to punish Netanyahu for allegedly violating a conflict of interest agreement meant to prevent him from dealing with the country's judiciary while he is on trial for corruption.
Netanyahu is barred by the country's attorney general from directly dealing with his government's plan to overhaul the judiciary, instead, Justice Minister Yariv Levin, a close confidant of Netanyahu, is spearheading the overhaul.
But on Thursday, after parliament passed a law making it harder to remove a sitting prime minister, Netanyahu said he was unshackled from the attorney general's decision and vowed to wade into the crisis and "mend the rift" in the nation.
That declaration prompted the attorney general, Gali Baharav-Miara, to warn that Netanyahu was breaking his conflict of interest agreement by entering the fray.
The Movement for Quality Government, a fierce opponent of the overhaul, asked the court on Sunday to force Netanyahu to obey the law and sanction him either with a fine or prison time for not doing so. It said he was not above the law.
"A prime minister who doesn't obey the court and the provisions of the law is privileged and an anarchist," said Eliad Shraga, the head of the group, echoing language used by Netanyahu and his allies against protesters opposed to the overhaul.
The prime minister responded saying the appeal should be dismissed and said that the Supreme Court didn't have grounds to intervene.
The Movement for Quality Government said the court had given Netanyahu and the attorney general a week to respond.
Netanyahu is on trial for charges of fraud, breach of trust and accepting bribes in three separate affairs involving wealthy associates and powerful media moguls. He denies wrongdoing and dismisses critics who say he will try to seek an escape route from the charges through the legal overhaul.
The overhaul will give the government control over who becomes a judge and limit judicial review over government decisions and legislation. Netanyahu and his allies say the plan will restore a balance between the judicial and executive branches and rein in what they see as an interventionist court with liberal sympathies.
Critics say the plan upends Israel's fragile system of checks and balances and pushes Israel down a path toward autocracy.
