Mr Lunney, a 50-year-old father of six, was left with life-changing injuries after being bundled into the boot of a vehicle outside his Derrylin home on September 17 and subjected to several hours of brutal attack.
The man, in his 50s, reportedly took ill when police were searching his home in Derbyshire, England this morning.
Taoiseach Leo Varadkar has met with the five directors of Quinn Industrial Holdings.
He was dumped on a road in County Cavan, Republic of Ireland, 22 miles (35km) from where he was abducted.More news: ATP Finals: Dominic Thiem beats Roger Federer on opening night in London
In particular, I wanted to thank Kevin Lunney for the resilience he has shown following his barbaric abduction, assault and torture.
He had the letters "QIH" cut into his chest with a Stanley knife, and told BBC NI's Spotlight programme this week that he feared that he would never see his wife and children again.
THE convicted criminal who died suddenly when police raided a house in Derbyshire is believed to be the "boss" rung by Kevin Lunney's abductors during his abduction.
The raid was part of a major operation across Ireland and the United Kingdom into the kidnap and torture of the Quinn Industrial Holdings director.More news: Watch this adorable beluga whale play fetch with a football
PSNI Detective Chief Inspector Julie Mullan said: "This was a truly horrific crime and we continue to work closely with our colleagues in An Garda Siochana and now also Derbyshire Constabulary to try and bring the perpetrators to justice".
Police on both sides of the border are under pressure to get results, following a long-running series of threats and other incidents.
"I am pleased that 150 additional officers have been assigned to the region in the last two years following a hard period for policing arising from the 2010 closure of Templemore College", he said.
The attack came amid five years of co-ordinated intimidation of businesses which own companies formerly run by Irish business tycoon, Sean Quinn.More news: Trump to pursue higher sales age for e-cigarettes
The Quinn family has condemned and distanced itself from those attacking the new owners.