The dissident republican group, the Real IRA, which has threatened to kill more PSNI officers, has refused an offer from a Londonderry priest to hold talks with him.
The group said police officers would be targeted "regardless of their religion, culture background or motivation".
The threat was made in a statement at a rally in Derry on Monday.
Fr Michael Canny said he hoped talks would happen in the future.
"I heard through a source that I would consider to be reliable on Wednesday evening that they didn't want to meet," he said.
"I certainly wasn't surprised but neither was I totally disappointed because the statement said 'at this time'.
"I consider that they obviously carefully considered what they were saying and the use of the phrase 'at this time' doesn't rule out the possibility of talking, not necessarily to me but to somebody else in the weeks or indeed in the months ahead.
"Door is open"
"While it was somewhat disappointing it was also hopeful."
Father Canny said his door "is open next week and the week after, and indeed anytime they so wish to use it".
"I would like to use the opportunity of talks to convince them that the only way they can obtain their goals is not through bomb or bullet, but rather through the democratic process and to talking not only to me and to other people, but also talking to the other people who are part of that equation, if they so wish to unite Ireland."
The priest said he had spoken to people in Rossville Street, William Street, and the surrounding area on Wednesday.
"Absolutely everybody that I have met was wholeheartedly supportive, and totally 100% behind the idea that someone has to talk to them, with a view to ending this campaign," he added.
Raymond McCartney from Sinn Fein said Fr Canny's offer had been a "genuine attempt" to engage in dialoge with the Real IRA.
Mr McCartney said it was time for dissident republicans to listen.
"I think increasingly they are becoming isolated within the community," he said.
"Who will talk to them? Well there have been many offers, they seem to be more willing to refuse than to take up the offers, but I think we have to keep at it.
"Now is the time to listen to what the people of Ireland are saying in overwhelming numbers."
The SDLP's Mark H Durkan said: "We are supportive of Fr Canny's attempts to talk to dissidents as we were supportive of Fr Patrick O'Kane's efforts."
"It is unfortunate, but not unexpected, that dissidents have rejected this approach. However at some point they must listen to the voice of the Irish people and stop this violence."
Monday's rally was organised by the 32 County Sovereignty Movement to mark the anniversary of the 1916 Easter Rising.
It is regarded as the political wing of the Real IRA, the organisation which claimed responsibility for the Omagh bomb which killed 29 people and unborn twins in 1998.
It also killed two soldiers in Antrim two years ago and exploded a car bomb in Derry last year.
In a statement read out at the City Cemetery in Derry, the Real IRA also expressed its opposition to the Queen's impending visit to Ireland.
"The Queen of England is wanted for war crimes in Ireland and is not wanted on Irish soil," the statement added.