Mr Finucane said: “Last night the PSNI came to my home to tell me of a threat by loyalists to attack me in my family home.
“I am committed to serving and representing all the people of this city and I will not be deterred from that by threats from anyone.
“This is not just a threat against me but it is also a threat against democracy.”
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Sinn Fein MLA Gerry Kelly said: “This reprehensible threat against John Finucane has to be condemned.
“The reference to his family home was clearly deliberate, given this is where his father was murdered by loyalists.
“It is a cowardly attempt to intimidate the newly-elected first citizen of Belfast. And while this must be a very traumatic experience for John and his family, I know that he will not be deterred from representing everyone as Ard Mhéara.”
A spokeperson for the PSNI said: “We do not discuss the security of individuals and no inference should be drawn from this. However, if we receive information that a person’s life may be at risk, we will inform them accordingly. We never ignore anything which may put an individual at risk.”