Both sides are unseeded for the European play-off draw at FIFA headquarters in Zurich, so can draw Switzerland, Italy, Croatia or Denmark, who are seeded as a result of their world rankings.
The two-legged ties, which can be determined by away goals and penalties, are to be played between November 9 and 14, with Italy the opponents the Irish teams will be hoping to avoid.
The eight best second-placed finishers from the nine European qualifying groups advanced to the play-offs, with Slovakia missing out after finishing runners-up to England. The four play-off winners will take the number of European teams at next summer’s finals to 14.
Russia qualified automatically as hosts while France, Portugal, Germany, Serbia, Poland, England, Spain, Belgium and Iceland advanced after topping their groups.
The Republic beat Wales to a play-off place and will be hoping to continue their recent run of success in the lottery format.
Two of the Republic’s three wins in eight play-offs have come in the last two attempts, but they have also experienced heartache in the format, including in the bid to qualify for the 2010 World Cup, when Thierry Henry’s handball helped France to advance.
Martin O’Neill’s men advanced to Euro 2016 at the expense of Bosnia and Herzegovina in the play-offs, but Northern Ireland are new to the format after qualifying automatically for the European Championship finals.
England and the Republic have risen in this month’s FIFA’s world rankings, while Scotland climbed too despite missing out on World Cup qualification.
England climbed three places to 12th in the rankings led by World Cup holders Germany.
Gareth Southgate’s side are the leading home nation after Wales, who, like Scotland, missed out on the finals, fell a place to 14th.
The Republic rose eight places to 26th, while Scotland climbed 14 places to 29th despite missing out on a place in Russia.
Northern Ireland fell three places to 23rd despite advancing to the play-offs.
Brazil are second in the rankings and Portugal third, with Argentina fourth, Belgium fifth and Poland sixth in an unchanged top six.