The 25-year-old paceman missed the last two Tests of the 3-0 whitewash of New Zealand earlier this season with a foot injury.
But he came through unscathed for MCC in their match against the West Indies earlier this week.
Jones and James Anderson appear to be vying for one spot in the final eleven come the start of Thursday's first Test.
But both quicks could play if Anderson's Lancashire teammate Andrew Flintoff is unable to bowl and the selectors decide they want four quick bowlers in the team.
All-rounder Flintoff has a bone spur in his left ankle and although he returned during the recent One-day series he was unable to bowl.
There has been speculation that England could play the hard-hitting Flintoff purely as a specialist batsman - which is what happened in the One-dayers.
England's chairman of selectors, David Graveney, said on Sunday: "We will be closely monitoring Andrew Flintoff's fitness in the run-up to Thursday.
"We feel that the more time he has to recover and receive treatment, the greater his chances of taking part in the game as an all-rounder."
Graveney was equally upbeat about the prospects of Surrey batsman Mark Butcher playing in his 43rd consecutive Test despite tearing a thigh muscle nine days ago.
The left-hander was due to prove his fitness in a One-day game for Surrey on Sunday but has now been withdrawn.
"We are confident that Mark Butcher will be fit and available for selection and the decision not to play him in Surrey's Totesport League match today (Sunday) is purely precautionary," Graveney added.
Kent's Robert Key, the first batsman to 1,000 first-class runs in England this season, is included as batting cover.
But his seam bowling County colleague Martin Saggers, who played in the second and third Test wins against New Zealand, has been omited now that Jones and Anderson are both fit again.
Earlier this year England enjoyed a 3-0 Test series win away to the West Indies, their first Test series victory in the Caribbean for 34 years.
Michael Vaughan (Yorkshire, capt), James Anderson (Lancashire), Mark Butcher (Surrey), Andrew Flintoff (Lancashire), Ashley Giles (Warwickshire), Stephen Harmison (Durham), Matthew Hoggard (Yorkshire), Geraint Jones (Kent, wkt), Simon Jones (Glamorgan), Robert Key (Kent), Andrew Strauss (Middlesex), Graham Thorpe (Surrey), Marcus Trescothick (Somerset)