The 24-year-old has returned to Manchester United after complaining of a sore calf during training Thursday, and will be unavailable for selection Friday and Tuesday.
A statement on the FA's website states that Cleverley "complained of a sore calf after training on Thursday afternoon" and later underwent a scan.
He has been ruled out for the two crucial matches at Wembley, and has returned to United for further treatment with the extent of the injury unknown.
England sits top of UEFA qualifying Group H, one point ahead of Ukraine and Montenegro, and two points ahead of Poland. The Three Lions host Montenegro on Friday while Ukraine entertains Poland in Kharkiv. On Tuesday, the Poles travel to Wembley while Ukraine plays lowly San Marino.