Trippier was surprisingly left out of Newcastle's team for last Saturday's Premier League opener against Southampton.
Tino Livramento took Trippier's place at right-back, while Brazil midfielder Bruno Guimaraes replaced him as Newcastle captain.
Reports suggest Trippier, who turns 34 next month, wants to quit Tyneside before the transfer window shuts on August 30.
Yet Howe is keen to retain the services of a player who was a regular during England's run to the Euro 2024 final.
"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," Howe told reporters on Friday.
Trippier joined the Magpies from Atletico Madrid in January 2022 in one of the first major signings by the club's Saudi Arabian owners.
He was key to the resurgence which saw Newcastle reach the 2023 League Cup final and play Champions League football last season.
However, Newcastle struggled to a disappointing seventh-place finish in the Premier League last term.
"He's played a huge role in changing the narrative around the team. When he first came in, we were in a very difficult position and it's easy to forget that," Howe said.
"He took a big risk to leave a huge football club and was having a successful period with them to come to a team battling relegation.
"Our relationship with him was pivotal and he galvanised the squad from the first day. His performances on the pitch also spoke volumes."
Trippier's absence from the team last weekend raised eyebrows, but ahead of Sunday's trip to Bournemouth, Howe said he remains a valued member of the squad.
"I'm surprised at a lot of the stories that have come out. He's a valuable member of the squad. His professionalism has been first class," Howe said.
"He's trained really well since he's come back from the Euros. He's only had two weeks training, hence the decision to not start him at the weekend and Tino has had a very good pre-season. He continues to be really valued by all of us."
