Ireland captain William Porterfield said the country deserves a chance to play Test cricket.
Just last week, the International Cricket Council announced a new process for associate nations who want to become full members.
Ireland's application is likely to be heard next month.
Before that, they have a two-match 50-over series with England - Porterfield told Sky Sports News that has to be their focus.
“From our point of view, over the next few days we need to put in two big performances here and come away with a couple of wins and whatever happens from there, happens.”
Ireland play England in Bristol tomorrow followed by a second ODI at Lord’s on Sunday.