The former England international has decided to leave Meadow Lane after playing only one game for the Magpies, having signed a five-year-deal with the League Two club.
"We're disappointed Sol has decided, from yesterday, he wants to end his contract," he told Sky Sports News.
"It's one of those things, he joined five weeks ago on a five-year contract as part of the long-term project, but ultimately he's come to the point, having been here five weeks, where he has decided it's not for him.
"Sometimes decisions are made, people make wrong decisions and Sol has decided this is not where he wants to be now.
"We're not going to kick up a fuss about it. If he wants to go, he needs to go and we genuinely wish him all the best."
Asked about the reasons Campbell gave for leaving, Trembling added: "Not an awful lot, just that this was not working out for him.
"I spoke to his representative this morning (Wednesday) and he has just decided it's not working out and he wants to move on. Perhaps things are not happening as quickly as he thought they might, but this is a five-year project, not a five-week project.
"We can't just become a Premier League club overnight. That is how it is. If he doesn't want to play here, we don't want him."