Barca heads into the first Clasico of the season on the back of victories over Valencia and Dynamo Kiev in LaLiga and the UEFA Champions League respectively.
It is the first time this season the Catalan side has won successive games and has eased the pressure on Koeman, who has the public backing of president Joan Laporta.
But Koeman's side is paying for a slow start to the campaign as it sits seventh on the league table, two points behind Madrid and five off leader Real Sociedad with a game in hand.
The Dutchman is no stranger to arguably the most famous fixture in football, having enjoyed a hugely successful stint at Barca as a player, and is aware of the magnitude of Monday's (AEDT) clash.
"I'm not under any more pressure. Being a Barca coach, I know the importance of what is ahead," he said at a pre-match news conference.
"I have lost and won Clasicos as a player. It is my first as a coach with an audience, which makes it different. I'm hopeful of winning the game.
"Winning is important as it would help us continue to build confidence with all the changes we made and the youngsters that are emerging.
"We work together to improve on the field. I don't feel like this is an exam for us, but it is a Clasico and we have to prove ourselves."
Laporta said ahead of the 2-0 loss to Atletico Madrid on 3 October (AEDT) that Koeman deserved time to turn things around in what is just his second season at Camp Nou.
Asked if he still feels the support of his president, Koeman said: "I don't have to be told every day that I will continue in the job.
"I feel supported but I know everything depends on results. It's not about a single game, though this is an important week ahead of us and we must be prepared.
Barca has not won any of its past four LaLiga games against Madrid and has lost the past three Clasicos in all competitions – not since 1965 has it lost more successively.
The Catalan side could find itself five points behind Madrid with a defeat, while a victory would see it leapfrog its bitter rival.
"We enter the game in a good mood thanks to two victories," Koeman said. "There is a lot of enthusiasm to play and show our fans we want to win.
"A win in a Clasico can make a lot of difference between the good or bad. We know it's one of the most important games, but there is a long season still ahead.
"We can show we are on the right track, making a positive result very important. We have to play without fear and show our desire to win from the start."
While results have improved for Barca, Madrid has collected just one point from its past two league matches and risks losing successive LaLiga games for the first time since 2018-2019.
"I'm a Barca coach, so it is difficult to answer, but I don't see a clear favourite," Koeman said when asked if there is anything to separate the two heavyweights.
"There is a difference in terms of experience, but we also have some experienced players in our squad, while they have some young players, so we are more or less the same.
"There are areas in which they are better than us and others in which we are better than them. We are playing at home and can have no fear. We want to show we're a good team."
Koeman confirmed Jordi Alba is available for selection, despite sustaining a knock against Dynamo Kiev, but Pedri, Ronald Araujo, Martin Braithwaite and Ousmane Dembele are not part of the squad.