Sherwood only took charge of Tottenham on a permanent basis in December following the departure of Andre Villas-Boas, but he has already come under considerable pressure following a poor run of results.
Sunday's 1-0 derby defeat to Arsenal followed a 4-0 thrashing at Chelsea and a 3-1 loss in the first leg of a UEFA Europa League last 16 tie against Benfica.
Yet although Tottenham, who sit fifth in the Premier League, are facing the prospect of exiting European competition and missing out on a top-four finish, Rose insists that he and his fellow players are fully behind Sherwood.
"You look at the best teams and they do well when they do have a manager to work with for a long time. Everyone loves Tim and the coaching staff," Rose is quoted as saying in the English media.
"You don't know what’s going to happen. We just want the fans to stick by us and obviously we would like a bit of continuity with a manager and we'll all be fighting to put it right, to get the top four maybe next season."
The forthright Sherwood has frequently cut an animated figure on the touchline since succeeding Villas-Boas.
And Rose added: "It's worrying when he's coming for you at half-time or full-time but no, it's great.
"You can tell that he's passionate and he loves the club. He knows the club from the first team down to the Under-9s.
"Once the manager took over the job in December, we were on fire. We won a lot of games and we've just had a bad week where we've just detached ourselves away from the top four."
Assessing Tottenham's campaign as a whole, Rose continued: "Individually, we might not have done so well this season and, as a team, we definitely haven't done that well. Some of the games we have lost at home and drawn at home, it's been unacceptable.
"We're all definitely disappointed that it looks like this season might end without us finishing (in the) top four."
