The Premier League champions host United at Stamford Bridge on Sunday afternoon (5 November) and could be 12 points behind Manchester City by kick-off, should Pep Guardiola's men beat Arsenal earlier in the day.
"Manchester City are already nine points ahead of us, Manchester United four and Tottenham one," Courtois was quoted as saying by Sky Sports.
"We must win games like these to remain at the top and close to the other ones. We will try to get the three points and will give our lives for it on Sunday."
Chelsea have lost just once in their last nine matches in all competitions against United at Stamford Bridge but arrive into Sunday's clash on the back of a heavy defeat in Rome, which prompted Antonio Conte to urge his side to "dig deeper".
The 3-0 loss the Blues suffered on Tuesday (31 October), was their first defeat in the Champions League this season and the third time they have conceded three goals in a game this term, having done so only twice in the whole of last season.
Chelsea's 1-0 win away at Bournemouth on Saturday was the only time they have kept a clean sheet since September, a remarkable change from last season, when the Blues boasted the league's third best defensive record.
"We have to react on Sunday as the Roma defeat was a bad image of ourselves", Courtois added. "We have to find that balance again and rediscover that mentality of not conceding goals and keeping clean sheets."
So far this season, Chelsea have won just two of their five Premier League games on home turf and have already lost two, as many as they did in the whole of last season, when they won 17 out of 19 matches at Stamford Bridge.
However, Conte can draw some comfort from United's dismal record in West London, where they have failed to score in three of their last four visits in all competitions. The 20-time champions of England have not won at Stamford Bridge in the Premier League since Javier Hernandez sealed a 3-2 win against nine-man Chelsea in October 2012 and Jose Mourinho has only won once in three visits to his old stomping ground, with Inter in 2010.
The Portuguese faced stern criticism for the way he approached their trip to Anfield two weeks ago, when they looked content to settle for a 0-0 draw. In Mourinho's first 18 months at Old Trafford, United have not won away from home against any of the so-called Big Six teams and have only scored once in six games.