Giovanni Trapattoni’s men, who sealed second place in Group B with a 2-1 win over Armenia on Tuesday, will either face Turkey, Bosnia and Herzegovina, Montenegro or Estonia for a place at next summer’s finals.
The Republic will also have the supposed advantage of playing the second leg at home. The first leg of the play-offs will be played on 11 and 12 November, with the return legs on 15 November.
The other seeds for the draw, which takes place on Thursday in Krakow at 1200BST, are Croatia, Portugal and the Czech Republic.
Already through to the finals, alongside co-hosts Poland and Ukraine, are holders Spain, Germany, Russia, Italy, France, the Netherlands, Greece, England, Denmark and best runners-up Sweden.
The finals draw is on 2 December in Kyiv and the tournament runs from 8 June until 1 July next year in eight venues.
Pot 1 (seeded teams)
Croatia (coefficient 32.723)
Republic of Ireland (28.203)
Czech Republic (27.982)
Pot 2 (unseeded teams)
Bosnia and Herzegovina (27.199)
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