According to Sky Sports, the Premier League clubs have agreed a season-long deal.
Chambers has been a fringe figure since joining from Southampton for £16m in 2014.
Arsene Wenger believes the 21-year-old's future lies in central defence and wants him to get regular action.
He started the opening-day defeat against Liverpool due to injuries but is behind new signing Rob Holding in the pecking order.
The imminent arrival of Germany international Shkodran Mustafi is likely to speed up his departure.
Wenger said last week that the signing of Mustafi would not affect Chambers' long-term future but confirmed a loan move is on the cards.
He said: “Future? No. The fact that he could go out somewhere and play? Yes.”
Aitor Karanka's Middlesbrough have started the season in fine form, picking up five points from their opening three league games.
