Sherwood was installed as Spurs boss on an 18-month contract following Andre Villas-Boas' axe in December, but other names have been linked with his post.
Departing Holland manager Louis van Gaal has been heavily touted with the White Hart Lane hotseat, but Rose has stressed that the dressing room "loves" playing under Sherwood and has urged the club and the fans to stick with him over the long run.
He told reporters: "We want the fans and everyone to stick by us and we would like a bit of continuity with the manager. As players we know as much as you about what's going to happen with the manager but you look at the best teams, they do well when they have a manager to work with for a long time.
"We will have to wait and see what happens but everyone loves Tim and the coaching staff. Tim is passionate, he loves the club. He knows it from top to bottom, from the first team down to the Under-9s.
"You know he will give 100 per cent for the club. He has played for the club, he knows what the games are about and what the fans expect from the team. It has been great working for him. I have known him for a long time and if it wasn't for him, I am not sure where I would be playing right now. He has helped me change my Tottenham career around, so I have got a lot to thank him for."