The measures were announced immediately after France introduced a state of emergency after the November 13th shootings and suicide bombings in which 130 people were killed in Paris.
Since then, more than 300 people have been placed under house arrest and three mosques have been shut for promoting extremism, an unprecedented step in France.
Parliament voted overwhelmingly in favour of extending the state of emergency for three months.
Till now, the measure, which gives police the power to carry out raids and arrests without judicial oversight, did not feature in the constitution, only in a separate law that has rarely been used.
A more concrete change proposed by the government would see, for example, powers of house arrest extended for six months after a state of emergency expires.
"There are three possibilities: the first one would be to write the state of emergency within the Constitution, in article 36, almost for aesthetical reasons, just to make sure we haven't forgotten anything. That would not change anything. The second one would have further guarantees written within the state of emergency, meaning that we would put into the Constitution some freedom guarantees that would allow the state of emergency to be closer to the normal rule of law, non-exceptional. Then the last possibility, that I think would be dangerous, would allow the state of emergency to be written in the Constitution in order to raise a number of freedom restrictions to a constitutional level... for example, new powers would be given to the police," Jacques Toubon, a Defender of Rights and former minister of Justice, told RFI.
"When it was first put forward, the Council of the State gave a negative review, saying that if the threat of terrorism was so high, the government should simply have another state of emergency rather than setting it in stone in the Constitution. That's what we need to look for here, will that mean the implicit creation of a permanent state of crisis that would have long-term and permanent freedom restrictions... and that would naturally be completely different from the state of emergency that is only in place for a given period, that has a beginning and an end. But if it becomes permanent, we no longer will be within the rule of law."
But others believe the state of emergency has to be written in the Constitution.
"I think the state of emergency, which is a state of exception, needs a constitutional basis, which it doesn't have yet, because it's just provided by a normal status, the status of 1955," Jean Philippe Derosier, an expert in constitutional law at the University of Rouen told RFI.
"Putting it into the Constitution makes sense for any measure that can be taken on the basis of this state of emergency and that may go against constitutional provisions... With a constitutional basis, then there is a balance between this state of emergency, the state of exception and the other rights and freedom guaranteed by the Constitution."
The state of emergency itself has been criticised by many in France.
Since it authorises raids on any location, day or night, including the homes of lawyers, journalists, and politicians, the fact that it that could be written into the Constitution worries many French people.
Climate activists have recently been placed under house arrest to prevent them from protesting during the UN talks outside Paris. Their lawyers have accused the government of abusing its power.
"If French people want to live in a State which abids by the rule of law, they should be very worried. Because as soon as the state of emergency is provided for in the Constitution, the government and the police will have extensive powers," Catherine Haguenaud-Moizard, a law professor at the university of Strasbourg told RFI.
"As long as Parliament adopts the state of emergency and they can prolong it for a number of months... France will be living in a kind of permanent state of emergency, or quasi permanent... I don't think it's a matter to be disregarded."
She added that compared to other countries, France was taking things quite far.
"France is going further than most countries, most democratic countries at least. For example the UK in 2001 had adopted a very very strong act of Parliament against terrorism but then the House of Lords and the European court of human rights had declared that this piece of legislation was contrary to the European convention of human rights and they changed it. Now, the Home secretary has wide powers to make decisions against suspected terrorists but it has to be authorised by a judge."
She gave another example: "In Germany, it's not possible to take all sorts of measures without the authorisation of a judge. In most democratic countries, nobody goes as far as France. France is going almsot as far as Egypt, which is not a democratic country by all regards. It's very worrying."
There were talks about stripping dual nationals of their French nationality, but the government has done a U-turn on that particular amendment.
The Elysée hasn't confirmed it is to drop it yet, but it has certainly been the subject of controversy here in France.
The right to remove the citizenship of anyone with another nationality was also among the proposed changes to the constitution, in cases where an individual was found guilty of acts that "constitute an attack on the fundamental interests of the nation or... an act of terrorism".
Many think that the government reacted too hastily after the attacks.
Jacques Toubon said adding this to the French Constitution would greatly endanger the French Republique.
"Stripping a dual national, who was born French, that measure would go against fundamental principles of the State, and the Council of the State also went against it... French nationality can't be seen as having several levels, nor categories... And by imposing this, we would create a new category of citizen whose nationality would be questionable," said Toubon.
"And it would lead to the suppression of the Jus Soli - meaning the right of anyone born in the territory of a state to hold that nationality... You cannot divide citizenship, and this measure would be dangerous and would go against everything we stand for in France."
Many politicians, from both right and left parties, have raised concern about what they say could be the temptation to slide towards an authoritarian government.