And Middlesbrough are on alert over Gayle, with the striker linked with a move to Teesside.
Chris Wilder is reshaping his Boro squad ahead of another season in the Championship, with the club targeting promotion back to the top flight.
The Daily Mail report that the trio have been told to train at the club’s academy ahead of expected summer moves.
Striker Gayle, 32, rarely featured under Howe last season, while midfielder Hendrick spent the second half of last season on loan at Queens Park Rangers. Defender Clark was left out of the club's 25-man Premier League squad in January along with Isaac Hayden and Jamal Lewis.
Gayle and Hendrick, 30, have two years left on their deals, while 32-year-old Clark is in the final year of his contract. Gayle and Clark joined Newcastle in the summer of 2016 from Crystal Palace and Aston Villa respectively – and helped the club win promotion to the Premier League.
Howe praised Gayle’s attitude late last season. United’s head coach said: “You have your players that are playing, but the guys that aren’t playing, behind the scenes, are so important to the culture and feel of the group.
"Dwight’s been pivotal to keeping that very, very positive. He’s been a big reason why the team’s done so well.”
The Teesside Gazette report: "Newcastle United have offered Middlesbrough the chance to sign striker Dwight Gayle this summer as they look to offload the striker.
"With his family settled in the North East, it is understood that the move to Boro would appeal to the player too.”
Asked about Clark in February, Howe said: “His attitude has been first class. I’ve got no problem with him at all. Unfortunately, due to numbers, he missed out,” he added.
Hendrick, 30, joined as a free agent two years ago during Steve Bruce’s time in charge after his contract at Burnley expired.
There’s understood Championship interest in all three players, though they may only initially leave the club on loan.