"There is no chance of the Euros for him. He will be back at the beginning of July," Wenger told reporters.
England face Russia, Slovakia and Wales in Group B of the 24-team tournament in France that starts on June 10.
Wenger said defender Per Mertesacker will also miss the final two games of the north London side's campaign with a hamstring problem, while midfielder Santi Cazorla will return to the squad after a five-month injury layoff.
"Cazorla came through well in a game in the reserve league and everybody else should be available. The season is over for Mertesacker due to his hamstring," Wenger said.
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Wenger admitted Arsenal lost an opportunity to end their 12-year wait for the Premier League title this season, after Leicester City were confirmed as champions on Monday.
Arsenal trail second-placed Tottenham Hotspur by three points and Wenger was eager to preserve his record of never finishing below their bitter rivals in his 20-year stay at the club.
"We have regrets because we had a chance to finish top of the league... Our target now is to secure our position in Champions League next year," the 66-year-old said.
"If possible to get in second place and at least secure third and on Sunday we can achieve that. If we win the two games we will be all right."
Captain Mikel Arteta is unlikely to be offered a new contract and Wenger thanked the 34-year-old midfielder for his contribution during his five-year stay at the Emirates Stadium.
"It is true he is out of contract (in the summer) and that his contract finishes it is a good opportunity for me to thank him for his contribution here, not only as a player but also as a leader," Wenger said.
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by John O'Brien/Sudipto Ganguly)
