Flutey and Shaw, who were both Test Lions in 2009, are struggling with calf injuries, while 2001 and 2005 Lion Wilkinson has complained of soreness in his shoulder.
Leeds Carnegie's Fourie has a bruised rib and Bath's Wilson is suffering from a sore back.
The quintet will continue to be treated and assessed by the England medical team in consultation with the medics from their clubs.
With the series opener against the All Blacks now less than two weeks away, Johnson has brought Bard Barritt, Dan-Ward Smith, Charlie Hodgson, Tom Wood and Joe Marler into his training squad.
Northampton flanker Wood, 20-year-old Harlequins prop Marler, Saracens centre Barritt and Wasps back-five forward Ward-Smith are all currently uncapped.
In contrast, Hodgson has made 31 Test appearances and toured New Zealand with the Lions five years ago.
England face New Zealand at Twickenham on Saturday, November 6 before taking on Australia, Samoa and South Africa on successive Saturdays.