Elliot - who started United's final three matches of last season and kept two clean sheets as the Magpies won the Sky Bet Championship title - is one of four goalkeepers in the party.
Murphy, who netted six times last season, is also part of a 37-man group for games against Mexico, Uruguay and Austria.
However Ciaran Clark, who missed Newcastle's win over Barnsley on the final day of the campaign, is unavailable due to a knee injury.
O'Neill will hold a three-day training camp in Cork next week before the full squad links up in Dublin on Sunday, 28th May.
The squad will then fly to Mexico to face Mexico on Thursday, 1st June before returning to Ireland to take on Uruguay (4th June) and Austria (11th June) at the Aviva Stadium.
The Republic of Ireland squad for the three games is: Colin Doyle (Bradford City), Rob Elliot (Newcastle United), Keiren Westwood (Sheffield Wednesday), Darren Randolph (West Ham United), Cyrus Christie, Richard Keogh, Alex Pearce (Derby County), Paul McShane (Reading), Shane Duffy (Brighton & Hove Albion), John O'Shea (Sunderland), Andy Boyle (Preston North End), John Egan (Brentford), Marc Wilson (West Bromwich Albion), Kevin Long, Stephen Ward (Burnley), Matt Doherty (Wolverhampton Wanderers), Aiden McGeady (Everton), Glenn Whelan (Stoke City), James McCarthy (Everton), Jeff Hendrick, Robbie Brady (Burnley), Harry Arter (Bournemouth), Conor Hourihane (Aston Villa), Stephen Gleeson (Birmingham City), Eunan O'Kane (Leeds United), Darron Gibson (Sunderland), Liam Kelly (Reading), Wes Hoolahan (Norwich City), James McClean (West Bromwich Albion), Callum O'Dowda (Bristol City), Daryl Horgan (Preston North End), Jonathan Hayes (Aberdeen), Kevin Doyle (Colorado Rapids), David McGoldrick (Ipswich Town), Daryl Murphy (Newcastle United), Shane Long (Southampton), Jonathan Walters (Stoke City), Adam Rooney (Aberdeen).