Sherwood only took charge of Spurs on a permanent basis in December following the departure of Andre Villas-Boas but 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 Europa League last-16 tie against Benfica but, although the north Londoners are facing an exit from Europe and a failure to reach the Premier League top four, Rose insists that they are behind the boss.
|22/1||Tottenham are 22/1 with Bet365 to beat Benfica 2-0|
"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 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."