Up to six families in north Belfast claim they have received threats from the UDA and have been told to leave the area.
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It is understood two families have now left the Tiger's Bay area.
A source told UTV that they personally approached members of the UDA to ask why the families had been targeted but the organisation says it has not issued any threats.
In a statement released to UTV, North Belfast UDA "totally refutes allegations that they have ordered people to move from their homes in the Tiger's Bay area".
"Times have changed in north Belfast, with the UDA now supporting community development and community relations initiatives as a means of developing and securing loyalist areas of north Belfast", the statement continued.
"We would appeal to anyone claiming to be intimidated from their homes in Tiger's Bay, or any other part of north Belfast, to contact police immediately and provide them with details."
Pastor Brian Madden, from the Elim Christian Centre in north Belfast, says he wants the leadership of the UDA to sit down with the families in question for talks.
He added that the threats have left people terrified.
"They're petrified, they're absolutely petrified", he told UTV.
"One of the families was in my home a few nights ago and the wife was very, very shaken after living 46 years in the community she can't believe this has happened to her.
"She's never ever been involved in criminal activity. She's never had her name taken by the police She's just totally and utterly shocked to the core".
Police in north Belfast say they are aware of local tensions in the Tiger's Bay Area and take the concerns of the community extremely seriously.
"While we do not discuss the security of individuals, we would never ignore nor fail to inform somebody of any information that might suggest that this individual might be at risk and should step up their personal security", Area Commander York Road, Chief Inspector Ian Campbell, said.
"We are working closely with community representatives across the board to discuss any concerns and work with them to address the ongoing issues. We will not tolerate any criminal activity, and where we have reports and evidence of any crimes being committed we will resolutely act to bring those responsible before the courts."
Police have appealed to anyone with a particular complaint or any relevant information to contact them.
© UTV News