The threats, issued by the dissident loyalist group, the South East Antrim Ulster Defence Association (UDA), is being taken seriously by the Police Service of Northern Ireland (PSNI) and by the media company, which also publishes the Belfast Telegraph, Sunday Independent, Irish Independent, the Herald and a range of regional titles.
Séamus Dooley, NUJ assistant general secretary, said:
"This is a vile attempt to intimidate editors, journalists and publishers.
"It is the latest in a series of threats in Northern Ireland against journalists but is all the more sinister because it is a blanket threat against two newspapers, titles which have served the people of Northern Ireland fearlessly and often in the face of threats.
"In a week in which we marked World Press Freedom Day this is a grim reminder of the threats faced by many journalists across the globe by those who have reason to fear a free, independent and questioning press.
IFJ General Secretary Anthony Bellanger said: “At a time when key workers such as journalists are working extra hours and in incredibly stressful situations this blanket threat against newspaper workers is sickening.
“The threat should be lifted immediately and unconditionally”.
The current threats are the latest in years of intimidation of the Sunday World and its journalists by loyalist paramilitaries. Sunday World journalist Martin O'Hagan, who was Secretary of Belfast and District branch of the NUJ, was killed in 2001.
He was shot dead by a gunman as he and his wife Marie walked home from a night out in a local pub in Lurgan on September 28, 2001. The shots were fired from a passing car, which drove away at high speed.
Mr Bellanger said: “These threats must be treated extremely seriously and action taken to protect the lives of journalists and newspaper workers. The IFJ stands in full solidarity with the workers and the NUJ in demanding action”.