Lille began negotiations for the talented full back with their French counter-parts last week, and after seeing an initial bid rejected, a deal has eventually been struck.
24-year-old Corchia has put pen to paper on a four-and-a-half year deal at Lille.
The news will come as a blow to Arsene Wenger, who had identified the former France Under-21 international as a potential long term successor to Bacary Sagna.
Sagna is expected to leave the Emirates at the end of the season, with the club no closer to agreeing a new contract with the full back.
Sagna is understood to want a three-year contract , which Arsenal are unwilling to offer to a player who turns 31 next month.
Sagna is likely to move abroad at the end of the season, although Manchester City have been linked with a move for the Frenchman in the last few days.
Wenger had hoped to sign Corchia in the summer, but will now be forced to turn his attentions elsewhere.