Provided that Group D favourites Switzerland do not slip up against either Georgia or Gibraltar, a win for either side at the Aviva Stadium would guarantee them automatic qualification.
Here, the PA news agency takes a look at some of the memorable games between the sides over the years.
Republic of Ireland 2 Denmark 1, World Cup qualifier, October 1956
Drawn in the same group as Denmark and England, Ireland launched their campaign in impressive style when goals from Dermot Curtis and George Gavin secured the points at Dalymount Park despite Aage Rou Jensen's late strike. They repeated the dose in the return leg,which they won 2-0, but a return of just one point from the double-header with England dashed their qualification hopes.
Republic of Ireland 1 Denmark 1, World Cup qualifier, October 1969
Having seen the initial game abandoned with the score at 1-1 as a result of fog a year earlier, Don Givens scored for the Republic to claim a draw when the sides reconvened in Dublin. However, it proved to be the only point they managed in the campaign.
Denmark 3 Republic of Ireland 3, European Championships qualifier, May 1978
Early goals from Frank Stapleton and Tony Grealish handed Ireland a dream start in Copenhagen, and Gerry Daly re-established their advantage after Henning Jensen had pulled one back. However, a Benny Nielsen penalty and Soren Lerby's piledriver ensured the spoils were shared.
Republic of Ireland 1 Denmark 4, World Cup qualifier, November 1985
Stapleton's sixth-minute opener at Lansdowne Road simply served to spark the Danes into life as Preben Elkjaer's double and further strikes from Michael Laudrup and John Sivabaek eased a talented Denmark side to victory. The game proved to be something of a watershed for Irish football with the departure of manager Eoin Hand paving the way for Jack Charlton's arrival.
Republic of Ireland 1 Denmark 5, World Cup qualifier, November 2017
A hard-fought 0-0 draw at the Parken Stadium three days earlier looked to have put the Republic in the driving seat with Russia beckoning, and Shane Duffy's sixth-minute header at the Aviva Stadium had them in dreamland. However after Andreas Christensen had levelled, Christian Eriksen's fine hat-trick and a late penalty from Nicklas Bendtner sent Age Hareide's men to the finals.
Denmark 1 Republic of Ireland 1, European Championships qualifier, June 2019
Ireland looked to be heading for defeat in Copenhagen courtesy of substitute Pierre-Emile Hojbjerg's glancing header, but Duffy once again found a way past keeper Kasper Schmeichel to power Alan Judge's late free-kick into the net and claim a valuable point.