The proceedings have been initiated in the High Court by two siblings who came on the scene in the direct aftermath of the shooting.
The Irish Independent understands the cases involve claims for nervous shock and allegations of negligence and breach of statutory duty by gardaí.
Mr Nkencho (27), a young Dubliner of Nigerian background, was shot outside his family home in Clonee on December 30, 2020.
Gardaí had been called to an incident nearby involving Mr Nkencho, who had allegedly assaulted a shop assistant and was reported to be brandishing a knife.
Armed support was called in after uniformed gardaí requested he drop the knife.
When the Armed Support Unit arrived they came across Mr Nkencho outside his house.
Gardaí said they felt threatened when he refused to put down the knife and were trying to ensure he did not get inside the house.
They said attempts at non-lethal methods of disarming him failed. An autopsy found he was shot six times.
His family said he had been suffering from mental health issues.
An inquest heard the cause of death was multiple gunshot wounds to the trunk.
However, the hearing was adjourned following an application from the Garda Síochána Ombudsman Commission, which is investigating the incident.
Members of Mr Nkencho’s family were in the house when the shooting occurred.
The Garda Commissioner, Minister for Justice and the Attorney General are named as defendants in the civil proceedings being taken by Mr Nkencho’s relatives.
Dublin law firm Harringtons LLP, which is representing relatives of Mr Nkencho, declined to comment on the lawsuits.
“In circumstances where proceedings are before the courts, neither this office nor my clients have any comments to make at this time,” said partner David Harrington.
One of the plaintiffs, the deceased man’s sister Gloria Nkencho, has previously called for a public inquiry and spoken of the impact on her family.
Speaking last year, she said she and her siblings had witnessed the most traumatic experience of their lives.
“This is an unimaginable pain I do not wish upon anyone. The death of my brother has caused a vacuum in my home. My mother described it as a wound that will never heal,” she said.
"My family are forced to relive the trauma when we walk past his room, walk down the stairs and see where we stood when he was shot and killed. We see the bullet holes in our windows.
"This shouldn't have happened. George did not deserve to die. He needed help, not bullets. My brother was a person, not an animal."
In an interview last year, Ms Nkencho said she tried to tell gardaí her brother had mental health issues and to get them to move away.
"I opened the door when I saw my brother and I said to the guards, 'He's sick, move'. I wasn't allowed past the entrance and I was pushed back into my house,” she said.