The England international became the first signing of the Saudi-backed era on Tyneside when he joined the Magpies in January 2022.
However, he is no longer a guaranteed starter for Howe's side this season and was an unused substitute in Newcastle's 1-0 victory over Southampton last weekend.
Additionally, Trippier was told last week that he will no longer be captaining the side in the absence of Jamaal Lascelles, with the armband being handed to Bruno Guimaraes.
The 33-year-old has made 92 appearances for the Magpies, scoring four goals and providing 21 assists in that time.
Howe responds to Trippier transfer speculation
Widespread reports have indicated that Trippier is open to a move away from St James' Park before the transfer window closes and Howe addressed the situation in his pre-match press conference earlier today.
Asked if the 33-year-old will remain at the club this season, the Newcastle boss said: "I hope so. I can never answer these questions with absolute certainty because it's football and the transfer window is always unpredictable. But my wish is that he stays here and continues to play for us.
"The situation is business as usual, Tripps has trained really well this week. I'm surprised by a lot of the stories that have come out," Howe added.
"[He's] a valuable member of the squad. His professionalism has been first-class. He continues to be really valued by all of us. His leadership skills, ability with the ball, his assist record, everything he brings to the group, is everything we want here."
Trippier has slipped behind Tino Livramento in the pecking order for the right-back role and has been linked with a move to Everton in recent days.
If Trippier were to complete a move to Goodison Park, it would see him reunite with Sean Dyche, who he played for during his time with Burnley.
Howe provides update on defensive transfer target
In recent days, Newcastle have been heavily linked with a move for Trippier's England teammate Marc Guehi, who is believed to be nearing an exit from Crystal Palace.
During the press conference, Howe was quizzed about the progress being made around signing the Palace defender.
He said: "No news. I'm a step removed from it, so it's always difficult for me to give you definitive answers, but I'm aware there's still work going on behind the scenes.
"I'm not getting regular updates, so what will be will be."
Guehi had an impressive Euro 2024 campaign with the Three Lions and is also on the radar of Liverpool, who are yet to make their first signing this summer.