Bell recently had his contract with the Magpies' League One rivals terminated by mutual consent.
The 29-year-old had been with the Sky Blues for the past three-and-a-half years after joining the club from Norwich.
Bell made 98 appearances and scored six goals during his time at Coventry, the last of which came against Notts at Meadow Lane on the final day of last season.
However, having struggled with injury, only seven of those outings came last season.
The Irishman has also represented Rushden & Diamonds, Luton and Leicester, where he was on loan, during a career spanning more than 350 games.
A statement from the Sky Blues read: "Coventry City would like to take the opportunity to wish David all the best in his future career."