David Alaba gave the visitors the lead with a stunning strike before Lucas Vazquez doubled Los Blancos' lead in the third minute of stoppage time. Sergio Aguero pulled one back in the 97th minute, but it was nothing more than a consolation goal for the home side.
Barca are currently placed ninth on the LaLiga table after the same number of games and Koeman is under huge pressure. The Blaguarana boss was met by a large group of disgruntled fans after Sunday's defeat as he was driving away from the stadium.
Supporters surrounded his vehicle to show their frustration, with some banging on the bonnet of the Audi and insulting him while he was driving.
Fans have been calling for Koeman's head for some time now, though many online condemned the behaviour.
Barcelona also issued an official statement, writing: “FC Barcelona publicly condemns the violent and disdainful acts that our manager experienced when leaving the Camp Nou. The club will take security and disciplinary measures so that such unfortunate events do not happen again.”
Speaking after the game, Koeman admitted he was disappointed to lose but added he thought the game was finely balanced. "I am disappointed with the result but the team tried until the last second," he said post-match, as per The Mirror.
"We didn't create many clear opportunities. It was an even match but in the end the small details proved key. It is difficult to find positives from losing at home to Madrid, but I liked the attitude of the players and the fans. There is another game on Wednesday, the season is very long so we will keep fighting."