The Blues have gone four Premier League games since recording a win, a run which has seen them slip out of the Champions League places with London rivals Arsenal profiting.
Portugal international Bosingwa concedes missing out on the Champions League would be a massive blow, but says the manager has the full backing of everyone at the club and they will keep fighting for fourth spot.
"To finish outside of the top four at the end of the season will be a disaster, so we promise to carry on fighting and we need the support of the fans,” he told reporters.
"The dressing room has confidence in the coach. Villas-Boas is a good coach and he's demonstrated that with his success in Portugal.”
Arsene Wenger recently alluded to the Stamford Bridge club when describing the "crisis" in English football at the moment.
However, the 29-year-old full-back says the club are not in a crisis and they are confident of moving up the Premier League.
"This is not a crisis, but something from a bad run of results," he added. "I think we can improve and move up the Premier League this season."