Midfielder McClean and goalkeepers Randolph and Doyle have linked up with the squad after a three-day training camp in Fota Island near Cork.
McClean and and Randolph will add experience to a 19-man squad.
The game is followed by a friendly with Uruguay on 4 June and World Cup qualifier against Austria a week later.
Republic boss Martin O'Neill named a 37-strong panel for the three matches a fortnight ago and a selection of senior players will report for duty after the trip to the United States.
West Ham teenager Declan Rice was called up by O'Neill last week but he has dropped out following the training camp.
Burnley defender Kevin Long and Preston midfielder Alan Browne could make their Republic debuts against the Mexicans.
Meanwhile, Shane Duffy and Eunan O'Kane escaped injury in a road collision as they travelled to meet up with the squad on Sunday.
Brighton defender Duffy and Leeds midfielder O'Kane were en route to the team hotel when their vehicle was involved in the accident.
The Republic will return after the game in the USA for two games at the Aviva Stadium, including the crucial qualifier with Austria on 11 June.
O'Neill's side are unbeaten in their Group D campaign and lie second, level on points with leaders Serbia after five rounds of fixtures.
Republic squad for Mexico game
Goalkeepers: Colin Doyle, Darren Randolph
Defenders: Cyrus Christie, Alex Pearce, Richard Keogh, Shane Duffy, Andy Boyle, Kevin Long, John Egan
Midfielders: Eunan O'Kane, Alan Browne, Daryl Horgan, Conor Hourihane, Wes Hoolahan, Stephen Gleeson, Callum O'Dowda, James McClean
Forwards: David McGoldrick, Daryl Murphy
James McClean, Darren Randolph and Colin Doyle have been called into the Republic of Ireland squad for Friday's friendly against Mexico in New Jersey.