Cleverley had already been a doubt for the double-header after he was substituted with the problem during his side's 5-5 draw with West Brom on the final day of the season.
The FA confirmed on Sunday that the Manchester United midfielder has not joined up with the squad at St George's Park
Roy Hodgson called up Manchester City man Jack Rodwell earlier this week in case Cleverley could not regain his fitness, so the England manager will not be adding another name to his 21-man party.
With Steven Gerrard and Jack Wilshere already out through injury, Cleverley's absence restricts Hodgson's options in midfield even further.
Rodwell, Michael Carrick and Frank Lampard are the England manager's only three recognised central midfielders, although James Milner and Phil Jones can operate effectively in that position too.
Lampard, along with Chelsea team-mates Ashley Cole and Gary Cahill, returned early from the Blues' tour of the United States last week in order to keep them fresh for the game against the Republic on Wednesday and the glamour tie against Brazil in the Maracana four days later.
Despite that, Rodwell is likely to play some part in the two internationals - something which will undoubtedly boost the player's morale after his first season at Manchester City was blighted by injury.
Hodgson's options in the attacking department decreased by one when Andy Carroll was forced to withdraw due to a heel injury earlier this week.