The Bayern Munich forward is the only European-based player involved as Miguel Herrera's side face an American outfit missing DaMarcus Beasley and Michael Orozco
The pair's domestic club Puebla have refused a request to release the players for the game in Phoenix.
US head coach Klinsmann is less than impressed, but will not be calling up additional players to his squad, which is now comprised of 20 names.
"We are very disappointed in the decision by Puebla," he said.
"When the match was scheduled, it was with the spirit that both teams would have access to their domestic-based players for what is a very important game for those competing for spots in the World Cup.
"Both MLS and most clubs in Liga MX honoured that idea. While we understand their position, it's a huge opportunity lost for DaMarcus and Michael and doesn't contribute to the continuation of this respected rivalry."
With Wednesday's game the first of four friendlies for Klinsmann's side before their opening World Cup match against Ghana on June 16, the German knows that time is running out for his players to show their worth.
The US, who will be without all of their European-based stars with the exception of Bayern forward Green, were comfortable in qualifying for the tournament in Brazil later this year, finishing top of the final CONCACAF section with seven wins from 10 matches.
Klinsmann, who appointed compatriot Berti Vogts as a special advisor on Sunday, got the better of the Mexicans in qualifying, drawing 0-0 in the Azteca Stadium before goals from Eddie Johnson and Landon Donovan earned a 2-0 success at home.
After a mixed qualifying campaign in which Mexico failed to qualify automatically for the finals in Brazil, head coach Jose Manuel de la Torre lost his job, and what followed was a period of serious instability.
Both Luis Fernando Tena and Victor Manuel Vucetich came and went in a short space of time, with current boss Miguel Herrera being appointed on an interim basis for the intercontinental play-off with New Zealand.
Herrera was an immediate success as Mexico thrashed their opponents 9-3 over two legs and he was subsequently given the job on a permanent basis as the Mexican Football Federation looked to belatedly create a feeling of continuity ahead of the World Cup.
Herrera has named an 18-man party for Wednesday's friendly, comprised entirely of players from Mexico's domestic league, although Jose Juan Vazquez and Oribe Peralta have withdrawn over respective visa and personal problems.