The campaign group ADF International wants the UK to follow the European Parliament's lead on the issue.
Last month, politicians in Strasbourg unanimously passed a resolution recognising the systematic killing and persecution of religious minorities in the Middle East as a genocide.
The vote was the first time the body has recognized an ongoing conflict situation as a genocide.
The resolution goes further, to say that those who intentionally commit atrocities for ethnic or religious reasons should be brought to justice for violations against international law, crimes against humanity, and genocide.
Speaking to Premier about the need for the UK to take action, Paul Coleman from ADF International said: "They're [IS] intention is clear.
"In the propaganda and magazines that they publish, they are crystal clear that they are intending to wipe out Christianity and other minority religions from their area in the Middle East.
"It's clear under international law that what's taking place does meet this threshold of genocide. So far as this issue has been debated in parliament the UK government hasn't denied that it just refuses to take a position.
"What we'd like to see is recognition that what's taking place is a genocide. The UK plays a major role in global politics... it could very easily sponsor a resolution at the UN Security Council."
Legal definition of genocide:
Genocide is defined in Article 2 of the Convention on the Prevention and Punishment of the Crime of Genocide (1948) as "any of the following acts committed with intent to destroy, in whole or in part, a national, ethnical,
racial or religious group, as such: killing members of the group; causing serious bodily or mental harm to members of the group; deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part1; imposing measures intended to prevent births within the group; [and] forcibly transferring children of the group to another group."
The move by European politicians comes weeks after British parliamentarians wrote to the Prime Minister asking him to take similar action.
It was headed up by Rob Flello, who told Premier at the time: "Daesh (IS) are an evil cult who have unleashed a tide of death on Christians and other minority religions in the areas where they have seized control.
"We must send a clear and unequivocal message to them that eventually they will be held to account by the international community for their atrocities.
"We hope the Prime Minister will now act swiftly to encourage the United Nations to describe these killings as the orchestrated genocide they are."
The Government has continuously said it's not its job to define the persecution as a genocide.
Speaking in the House of Lords last week, the Earl Courtown said: "It is a long-standing government policy that any judgments on whether genocide has occurred are a matter for the international judicial system, rather than Governments or other non-judicial bodies.
"However, let me reassure noble Lords of the Government's commitment to defeating Daesh and preventing further abuses."
Listen to Premier's Marcus Jones speaking to Paul Coleman: