The Republic of Ireland want to arrange a friendly against England, which would be their first meeting since 1995.
The last match between the sides - at Dublin's Lansdowne Road - was abandoned after 27 minutes due to crowd trouble.
after their Euro 2012 first-leg play-off against Estonia and are keen to face high-profile teams before their likely finals appearance.
It is not yet certain if the match will be agreed before the draw for the finals is made on 2 December.
Football Association of Ireland (FAI) chief executive John Delaney told the Sunday Independent: "I think there is a real enthusiasm, particularly from the English FA, to have England in Dublin sooner rather than later."
Republic assistant manager Marco Tardelli was more focused on securing a place in the finals before considering the build-up to the event, which takes place in Poland and the Ukraine from 8 June to 1 July.
"For us, all teams are the same. I think in Europe, there are many strong teams," the Italian said.
"For us, if we play against Italy or England, it's no problem because we must play these games. The teams are not important, it is important to qualify."
England could also be Sweden's first opponents in their new national stadium when it opens next year in Stockholm.
The proposed friendly at the new 50,000-capacity arena is scheduled for November 2012.
It is understood that the fixture has been agreed as part of a reciprocal arrangement which brings Sweden to Wembley on Tuesday.
England also played the first match in Sweden's current national stadium, the Rasunda, in May 1937.
England won that game 4-0 but have not beaten Sweden in 12 games dating back to 1968.