Ireland captain Jason Molins wants his side to take on England in an annual one-day game.
Ireland Cricket Team
The Irish defeated West Indies by six wickets in Belfast on Thursday and have also beaten Surrey and Zimbabwe in the past 12 months.
"England play Wales as a warm-up to the NatWest Series, why not us as well?," Molins told BBC Sport.
"If we had Ed Joyce from Middlesex and Adrian McCoubrey of Essex, I'm sure we could give them a game."
Ireland are eager for more international exposure and Molins has previously called for them to be allowed to join Scotland in playing one-day league cricket against the English counties.
Peter Thompson, chief executive of the Irish Cricket Union, said the idea of a game against England had been canvassed with their counterparts at Lord's.
"We'd like to try and arrange more games against the Test nations," he told BBC Sport.
"We've written to the ECB about this but they say it isn't a possibility at the moment due to the amount of international cricket they play."
The ECB confirmed that England would not be playing Ireland in the foreseeable future.
"England's schedule is crammed, so they would not face Ireland this year or next," a spokesman said.
"But we would give due consideration to any proposals the Irish make.
Ireland will host the ICC Trophy next year, a tournament which offers five qualifying places for the 2007 World Cup.
[Edited on 19-6-2004 by reverse_swing]