McGuigan family members were told the threats are from "republican elements".
But police have declined to elaborate on whether they might be attached to any organisation.
BBC NI's Spotlight programme reveals that three members of the extended McGuigan family have received warnings.
Senior police officers have said current members of the Provisional IRA had a role in Mr McGuigan Sr's killing.
But Sinn Féin has said the IRA "has gone away".
BBC NI Spotlight reporter Darragh MacIntyre investigates the background to the political crisis, which was sparked by the murders of Mr McGuigan Sr and Gerard 'Jock' Davison in Belfast earlier this year.
The two men were once comrades who both killed for the Provisional IRA, and were part of an IRA front group called Direct Action Against Drugs.
But their connection ended in a bitter personal dispute.
When Mr Davison was shot dead near his home in May, some republicans concluded Mr McGuigan Sr was responsible.
The programme examines how the ground was laid for Mr McGuigan Sr's murder that took place outside his home four weeks ago - an attack that took on wider significance when the Police Service of Northern Ireland's Chief Constable George Hamilton said current members of the Provisional IRA were involved.
Former IRA member Anthony McIntyre, a frequent critic of Sinn Féin and who knew Mr McGuigan Sr, told the programme he believed IRA members were involved.
"When Kevin McGuigan was shot, it never once entered my mind it was anybody other than the IRA," he said.
"It was standard IRA modus operandi and it was a type of killing that the IRA carried out.
"The IRA certainly had a motive for killing Kevin McGuigan and that motive was that the IRA would need to dissuade or deter people who would want to have a pop at senior figures."
But Sinn Féin MLA Alex Maskey told the programme the IRA no longer exists.
The programme can also reveal one part of the IRA's command structure that republicans have said almost certainly continues to exist.
BBC NI Spotlight will be broadcast on BBC One Northern Ireland at 22:40 BST on Tuesday 8 September.
Relatives of Kevin McGuigan Sr, the republican allegedly murdered by members of the Provisional IRA, have been told by police that their lives are also under threat.