Wicket keeper-batsman Bairstow and off-spinning all-rounder Moeen were left out of the squad for the ongoing first Test against the West Indies.
England’s white-ball team might be without their key members including Joe Root and Ben Stokes as the third ODI is set to take place on August 4, the day before England play Pakistan in the first of a three-Test series at Manchester’s Old Trafford.
Paul Collingwood, England’s assistant coach will be the head coach for the series, with former England opener Marcus Trescothick in his backroom staff.
The England ODI side will go into an on-site training camp at the Rose Bowl on July 16 where they will play two intra-squad warm-up matches before the selectors pick a final squad.
The series will be played behind closed doors, starts on July 30, with remaining matches on August 1 and 4.
England training squad for Ireland ODIs:
Eoin Morgan, Moeen Ali, Jonny Bairstow, Tom Banton, Sam Billings, Henry Brookes, Brydon Carse, Tom Curran, Liam Dawson, Ben Duckett, Laurie Evans, Richard Gleeson, Lewis Gregory, Sam Hain, Tom Helm, Liam Livingstone, Saqib Mahmood, Matt Parkinson, Adil Rashid, Jason Roy, Phil Salt, Reece Topley, James Vince, David Willey.