The 32-year-old paceman is among four Lancashire players in the 24-man squad ahead of the three ODIs at Hampshire’s Ageas Bowl.
Liam Livingstone, Saqib Mahmood and Matthew Parkinson are also in the squad captained by Eoin Morgan.
All were among the training squad of 55 named in May, from which teams to represent England in all formats this season are to be selected.
Gleeson missed out on the Test squad for the current series against West Indies but always stood a better chance of his first international cap in one of the shorter formats.
The series in Southampton starts on July 30, with further matches on August 1 and 4.
The squad of 24 will go into camp at the Ageas Bowl next Thursday, will live and train on-site and will play in-house warm-up games on July 21 and 24.
Paul Collingwood has been named England’s head coach for the series. No players in the current Test team have been considered – a further Test series against Pakistan is due to start the day after the final ODI.
England ODI training squad: Eoin Morgan (Middlesex) captain, Moeen Ali (Worcestershire), Jonathan Bairstow (Yorkshire), Tom Banton (Somerset), Sam Billings (Kent), Henry Brookes (Warwickshire), Brydon Carse (Durham), Tom Curran (Surrey), Liam Dawson (Hampshire), Ben Duckett (Nottinghamshire), Laurie Evans (Sussex), Richard Gleeson (Lancashire), Lewis Gregory (Somerset), Sam Hain (Warwickshire), Tom Helm (Middlesex), Liam Livingstone (Lancashire), Saqib Mahmood (Lancashire), Matthew Parkinson (Lancashire), Adil Rashid (Yorkshire), Jason Roy (Surrey), Phil Salt (Sussex), Reece Topley (Surrey), James Vince (Hampshire), David Willey (Yorkshire).