Christian Solidarity Worldwide is calling upon the UK Government to raise concerns about the draft constitution and a controversial decree with the Egyptian government as a "matter of urgency".
Violent protests have been held daily since President Mohammed Morsi issued a decree giving himself sweeping powers and removing power from the judiciary to challenge his decisions.
Despite the unrest, the Egyptian government has signalled its intention to go ahead with a referendum on the draft constitution on 15 December.
The United Action for Egyptian Christians (UAFEC) has called upon the Egyptian President to cancel the decree, postpone the referendum and allow the judiciary to carry on with their responsibilities.
"We believe that unless a consensus is reached for the constitution the political and social turmoil will continue," the UAFEC said.
CSW said the draft constitution does not offer adequate protection for political or religious freedoms, or equal rights for women.
The organisation, which works for religious freedom, criticised the way in which the draft constitution was rushed through parliament without proper consultation after opposition parties and church representatives withdrew early on from the drafting process in protest.
Mervyn Thomas, CSW's Chief Executive, said, "We extend our heartfelt condolences to the families of those who died, and the many people injured in the violence in Cairo.
"We join UAFEC in calling on the Egyptian government to suspend the deeply divisive decree and constitution, and urge the government to uphold the right to peaceful protest and to prevent any further escalation in violence.
"We also call on the UK government to make urgent representations with the Government of Egypt regarding the need to reconsider both the decree and the draft constitution in the interests of national unity."
At least six people have been killed and 600 injured in demonstrations against the Muslim Brotherhood.
Bishop Angaelos, General Bishop of the Coptic Orthodox Church in the United Kingdom, said he was praying for those those who had been injured or or lost loved ones on the streets of Cairo in recent clashes.
"It is sad to see, after a strong national spirit became visible in Tahrir square almost two years ago, that Egyptians are now being attacked in the streets while living their newly found rights and expressing their opinions and demands," he said.
"There is no doubt that the issue of the draft constitution is one that is close to the hearts of many, and should thus be dealt with at this initial stage by consensus rather than simple majority, before being presented for referendum.
"Only then will Egyptians be able to move forward productively and embrace this new era of their contemporary history. We continue to pray for peace, safety, wisdom, restraint and protection for everyone in Egypt."