Tomkins, among 11 uncapped players included in Stuart Lancaster's original squad for next month's series in South America, was ruled out with an ankle problem while his Saracens team-mate Goode has been sidelined with a shoulder injury.
"Our commiserations go to Alex Goode and Joel Tomkins. It's disappointing that they have to miss out but they have both had excellent seasons and I am sure they will be back in contention next season," Lancaster said.
Uncapped Northampton Saints backs Luther Burrell, 25, and Stephen Myler, 28, have been drafted in as replacements.
With several senior players being rested for the tour, including regular captain Chris Robshaw, flanker Tom Wood will lead England when they play a combined South American team in Montevideo on June 2 and then Tests against Argentina in Salta (June 8) and Buenos Aires (June 15).