The one-off tie would slot in between the World Cup and the five-Test Ashes series against Australia, the Press Association reported.
The England and Wales Cricket Board’s cricket committee is now expected to review the proposal.
Ireland slipped to a five-wicket defeat to Pakistan in their maiden Test match last month. The International Cricket Council (ICC) approved four-day Tests in October.
England will host the World Cup, with the tournament ending on July 14, 2019, while the schedule for the Ashes series that follows is yet to be finalised.