Archer, who has been ruled out "for the rest of the summer", spent 17 months on the sidelines following elbow and back injuries before returning to international cricket in January.
He played five matches for Mumbai Indians before the nagging injury ended his Indian Premier League (IPL) stint last week.
"It has been a frustrating and upsetting period for Jofra Archer," Rob Key, managing director for England men's cricket, said in a statement.
"He was making good progress until a recurrence of the elbow injury, which kept him out for an extended period previously. I'm sure we will see Jofra back to his best and winning games for England, whatever the format. Hopefully, sooner rather than later.
Bairstow returned to the side after a freak golf injury in September that kept him out of the T20 World Cup in Australia.
Wicketkeeper Ben Foakes (30) was dropped to accommodate Bairstow for the four-day contest against Ireland at Lord's from June 1st.
"It was a seriously tough decision to leave Ben Foakes out of the squad," Key said.
"He has been excellent for England in the last year, but Jonny Bairstow is one of the best players in the world whose performances last summer epitomised what we are about."
Warwickshire seamer Chris Woakes (34) returns to the test squad for the first time since March 2022, while Durham fast bowler Mark Wood (33) is also back after skipping the tour of New Zealand.
The injuries suffered recently by skipper Ben Stokes (left knee) and seamer James Anderson (groin) did not appear serious, with both being named in the squad.
The Ashes series against Australia begins at Edgbaston on June 16th.
England squad: Ben Stokes (captain), James Anderson, Jonathan Bairstow, Stuart Broad, Harry Brook, Zak Crawley, Ben Duckett, Dan Lawrence, Jack Leach, Ollie Pope, Matthew Potts, Ollie Robinson, Joe Root, Chris Woakes, Mark Wood.
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