Hosts Bosnia come into the fixture on a three-game winning streak, while the Republic will be hoping to bounce back following defeat against Poland in their final group game.
Friday's hosts enter the playoff phase off the back of a vital three-game winning streak that saw them finish in third place in Group B behind Belgium and Wales.
Mehmed Bazdarevic's side produced a standout win against Wales at home in October in order to prop up their ambitions of making it to France, and the country's coach will be delighted at how his men then built on that win with a 3-2 away victory over Cyprus to ensure themselves a third-place finish.
Much of that work was done without the presence of Roma's Edin Dzeko through injury, but the frontman has now returned to league action where he has been in the goals for the Rome club in Serie A.
Bosnia have never actually made it into a European finals before, but they did make history two summers ago when they featured in the World Cup in Brazil.
Bazdarevic will be hoping to make use of the country's favourable home record, which has seen Bosnia unbeaten in four competitive matches, in order for them to build some sort of lead before they make the visit to the Aviva Stadium next Monday.
To gain a spot at next summer's showpiece, Herzegovina will need to improve their record in European playoff history, after they lost their sole opportunity when beaten 6-2 by Portugal as they sought to take a place in the 2012 run of the tournament.
Recent form: WWLWWW
Republic of Ireland
O'Neill's charges produced a standout comeback at the back-end of the qualification campaign in order to secure a berth in the playoffs.
The Republic had looked like an optimistic outside bet to edge into France next year, but they put together a memorable victory over world champions Germany to give themselves hope of qualification.
O'Neill's men might well have gained automatic progression into the finals had they not fallen short to a defeat in their final game against Poland, and the ROI will now be fully focused on making sure that the job gets done over two legs against Bosnia.
The Irish have some experience in the tense stages of the playoffs, given that they made their way into Euro 2012 under Giovanni Trapattoni as they defeated Estonia 5-1 over two legs.
The Republic have had to deal with a number of issues in the build-up to the game, however, and the country's coach will be keen to stress that his team draw on all of the experience that they have at their disposal come Friday.
As things stand, O'Neill is missing some key individuals through either suspension or injury, though the coach has been adamant in stating that his squad will not be influenced by the absences of John O'Shea and Jon Walters.
Recent form: DDWWWL
O'Neill will have to deal without O'Shea and Walters in the first leg due to suspension, while Shay Given and David McGoldrick are just two of a number of players to have dropped out of the squad because of injury.
The Republic will be sweating on the fitness of Shane Long, but he could make a return from an ankle issue in time to feature in Zenica.
As for the hosts, Bosnia are to be without Muhamed Besic who is serving a one-game ban, while Asmir Begovic, Anel Hadzic and Miralem Pjanic are part of a group of players all within one booking of a suspension.
Bazdarevic has included just one change of name to his squad for the fixtures, which has seen the youngster Rade Krunic come into the fray.
Bosnia-Herzegovina possible starting lineup:
Begovic; Mujdza, Spahic, Sunjic, Zukanovic; Pjanic, Hadzic, Sahilovic; Visca, Ibisevic, Dzeko
Republic of Ireland possible starting lineup:
Randolph; Coleman, Wilson, Keogh, Brady; Whelan, McCarthy, Hendrick, McClean: Keane, Long
Head To Head
The sides have only faced each other once before, where Ireland were able to rise as victors in a 1-0 win courtesy of Long's finish in a friendly in Dublin during 2012.
We say: Bosnia-Herzegovina 1-1 Republic of Ireland
Bosnia's comfortable victory over Wales shows that the country can often be a tough place to visit, while the home side will also be boosted by the possible return to international action of Dzeko up top. Ireland will by no means shy away from the challenge, however, and they will aim to bounce back from the defeat to Poland with a result that should leave them in a strong position going back to compete on home soil come Monday.