The Magpies have announced that the pair will both leave the club when their respective contracts expire at the end of the month.
Clark arrived from Aston Villa in the summer of 2016, and he went on to help Newcastle win promotion to the Premier League in his first season with the club.
The defender, who spent last season on loan with Sheffield United, made 129 appearances for the Magpies, chipping in with 11 goals and four assists.
Longstaff, meanwhile, worked his way through the academy ranks before making an instant impact on his Premier League debut, scoring the winner against Manchester United in October 2019.
The 23-year-old made a total of 20 appearances for Newcastle and has spent time on loan with Aberdeen, Mansfield Town and Colchester United.
Newcastle will continue to support Longstaff through his recovery from an anterior cruciate ligament that he suffered in December.
The Magpies have also confirmed that Harry Barclay, Niall Brookwell, Dan Langley, Joe Oliver, Josh Stewart and Isaac Westendorf will all leave the club at the end of their contracts.
Chris Wood will join Nottingham Forest on a permanent basis on July 1 after spending the second half of the season on loan at the City Ground.
As well as announcing a number of departures, Newcastle are set to retain the services of several members of Eddie Howe's squad.
The Magpies have taken up the option to extend the contracts of Matt Ritchie and Mark Gillespie by another year.
Dan Burn, who arrived from Brighton & Hove Albion in January 2022, has activated a one-year option on his deal, extending his stay until at least 2025.
Loris Karius and Paul Dummett, who are both out of contract at the end of June, have been offered new deals by the club.