The party leader is planning a series of visits to European countries in the run up to the Scottish Parliament elections in 2003.
Mr Swinney's trip across the Irish Sea is aimed at developing closer links between the two countries.
Monday's meeting with the Taoiseach comes the week before Mr Ahern makes a re-arranged visit to Scotland.
He had been due to travel to Lanarkshire in February for the unveiling of a monument to victims of the Irish famine at Carfin.
However, he cancelled the visit after Labour MP Frank Roy said its timing - immediately after a Rangers-Celtic match - could trigger sectarian violence.
The handling of the issue threatened to sour relations between Scotland and Ireland.
Both men will also be keen to put the last week behind them.
High profile visit
The SNP saw its share of the vote fall at the general election and it lost one of its Westminster seats to the Conservatives.
Mr Swinney - who has admitted that lessons must be learned from the campaign - wants to hit the ground running for the 2003 elections with a high profile visit.
He will also meet enterprise and tourism leaders in Dublin.
Meanwhile, Mr Ahern has lost a referendum which sought to ratify the Treaty of Nice, allowing the enlargement of Europe.
This was embarrassing for the leader of a country which has always been proud of its pro-European credentials.