Bairstow was given rest after the two-Test series against Sri Lanka, where he had scores of 47, 35 not out, 28 and 29.
Wood was also given a break after the series in the island nation where he went wicket-less in the first Test and took three wickets in the second.
"Moeen Ali will return to England for a break while Jonny Bairstow and Mark Wood join the squad having been rested for the first two Tests," the England and Wales Cricket Board (ECB) said in a statement.
England squad for Third Test:
Joe Root (C), James Anderson, Jofra Archer, Jonny Bairstow, Dominic Bess, Stuart Broad, Rory Burns, Zak Crawley, Ben Foakes, Dan Lawrence, Jack Leach, Ollie Pope, Dom Sibley, Ben Stokes, Olly Stone, Chris Woakes, Mark Wood.
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