The Scotland international appeared in nearly all of the Magpies league games last season, getting three goals and five assists in 35 games.
According to the Telegraph, Newcastle boss Rafa Benitez is willing to let the 28-year-old go for a fee of around £15m, as they look to recoup the £12m they paid for him two years ago.
Ritchie is no stranger to the Championship, having been a key part of Bournemouth's title-winning campaign in the 2014/2015 season, giving both Stoke and Middlesbrough hope that he can help them return to the Premier League.
Newcastle enjoyed a successful season, considering their limited finances and investment from owner Mike Ashley, finishing 10th but only 11 points above the relegation zone, and may be hoping that the money gained from selling Ritchie will enable them to improve the squad for next season.
The Scotsman has already been linked with a return to Bournemouth, with the Magpies reportedly interested in a swap deal for forward Josh King.
Middesbrough, however, have already shown that they are willing to spend large sums of money to improve their side, as they have recently paid a rumoured £10m for Bristol City's Aden Flint and Sunderland's Paddy McNair.
Tony Pulis, who will have to battle with his former side Stoke for Ritchie's signature, will be hoping that Middlesbrough can return to the Premier League at the second time of asking.
Stoke, however, will be disappointed with their relegation to the Championship, and will need to keep some of their key players if they wish to be promoted next season.