And the newspaper reports Ritchie will accept the Championship side's offer in the next few days, despite favouring a return to former club Bournemouth.
However, it is claimed the £15 million transfer could take longer to finalise, as Newcastle want to be sure of a replacement before sanctioning the Scotland international's exit.
Ritchie is likely to replace Xherdan Shaqiri at Stoke, with the sought-after Switzerland international now attracting interest from Manchester United, according to the Sun.
A £12 million marquee signing for Newcastle two years ago, Ritchie fired the Magpies to the Championship title in his first season at St James' Park, and will hope to have a similar impact at Stoke.
The relegated Potters have already spent close to £20 million on two new signings, Benik Afobe and Oghenekaro Peter Etebo, with Adam Federici also arriving on a free transfer.