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He also held talks with Crystal Palace, while Serie A side Roma also showed interest in signing the 24-year-old.
But the Gunners refused to do business with Roma because of the way they handled a potential move for defender Kostas Manolas earlier this summer.
The Cherries have also signed Romanian winger Mihai Dobre, 18, from FC Viitorul for their under-21 squad.
Wilshere has 34 international caps and played six times for England over the summer - including three games at Euro 2016 - but was not named in Sam Allardyce's first squad this week.
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He has started only 80 Premier League games for Arsenal in eight years, making two substitute appearances so far this season.
Wilshere is the second established England player to move on loan this week, with Manchester City goalkeeper Joe Hart moving to Italian side Torino.
Meanwhile, German side Werder Bremen have taken Arsenal midfielder Serge Gnabry, 21, on loan, while defender Calum Chambers, 21, has joined Middlesbrough on loan for the rest of the season.
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England midfielder Jack Wilshere has joined Bournemouth on a season-long loan from Arsenal.