Although Mr Donaldson could not say where the threat came from, Mr Poots has claimed that dissident republicans are responsible.
The warning was passed on to both DUP representatives by police, who told them that a threat had been made to them and their respective wives and children.
Mr Donaldson explained that the threat was very specific and "to the effect that we would all be shot if we didn't leave the area".
While he is "very concerned" by the situation, the MP added that he would not be leaving the area which he represents - nor would he be deterred from his usual parliamentary work.
"When it involves your family, it goes to a new low," Mr Donaldson said.
"As a politician who has been involved in the peace process now for many years, I suppose in a way I've had many threats made against me - but this is the first time that my family has been threatened in this way.
I think it is absolutely deplorable that someone would sink to such a low depth to threaten women and children.
Jeffrey Donaldson, DUP
Last week, it was revealed that death threats had been made against Alliance MP Naomi Long.
Her party has been repeatedly attacked, since a decision was taken by Belfast City Council to restrict the number of days the Union flag can be flown from City Hall.
Alliance holds the balance of power on the council and didn't give its support to a nationalist motion to remove the flag, instead tabling an amendment which meant it would fly on designated days.
Crowds have taken to the streets for nine consecutive nights after the ammended motion was passed, and violence has broken out at a number of protests across Northern Ireland.
But Mr Donaldson refused to call for an end to demonstrations.
"I believe that if people want to protest they have the right to do so, but they should do so peacefully and within the law," he said.
"Those who engage in acts of violence do enormous damage to the cause they claim to represent."
David Cameron addressed the issue of the disorder during Prime Minister's Questions in the House of Commons on Wednesday.
"In no way are these people being loyal or standing up for Britishness," he said.
"Violence is absolutely unjustified in those and in other circumstances."
Mr Cameron also told Naomi Long that she had the "complete solidarity" of her fellow Westminster MPs and praised her political work amid the intimidation.
The East Belfast MP welcomed a commitment from the Prime Minister to meet with her, and with Justice Minister David Ford, over the violence in Northern Ireland.
"This gratuitous violence, orchestrated in part by loyalist paramilitaries, is nothing less than an attack on democracy," Ms Long said, calling for strong leadership at Stormont and Westminster.
"My Alliance colleagues and I will not be intimidated from doing the job we are mandated to do, and that will remain the case."