Manchester City boss Manuel Pellegrini says he "did not want to" shake hands with Chelsea manager Jose Mourinho after the 2-1 defeat by the Blues.
Mourinho jumped into the crowd to celebrate after striker Fernando Torres scored his side's injury-time winner against City.
"I did not expect anything different from him," said Pellegrini. "It is normal."
Mourinho said: "If they feel I did something wrong, I apologise."
The Portuguese, 50, defended his actions by claiming he was trying to get to his son in the crowd when he celebrated the winning goal.
"This is the drama of the last minute," said Mourinho. "He [Pellegrini] lost the game in the last minute, and I have sympathy with that.
"In other countries and cultures we shake hands before the game, and that is it.
"If he was sad because of the result I understand, if he was sad because I did something wrong, I accept that too."
Mourinho succeeded Pellegrini as Real Madrid head coach after the Chilean was sacked in 2010, despite gaining a then record points total with the Spanish giants.
In 2011, Mourinho questioned both his predecessor's achievements at the Bernabeu and choice of club after Pellegrini subsequently decided to join Malaga.
"Second place is just the first loser," said the Portuguese ahead of a match between Malaga and Real. "If Madrid were to fire me, I wouldn't go to Malaga. I'd go to a top-level team in Italy or England."
Both managers played down their rivalry when questioned about it before the meeting on Sunday.
At Stamford Bridge, Chelsea had taken the lead through Andre Schurrle before Sergio Aguero equalised.
However, Torres pounced to score the winner after a mix-up between City centre-back Matija Nastasic and goalkeeper Joe Hart.
Pellegrini added that his side need to "erase these problems" if they want to challenge for the Premier League title.
City have lost three league games so far this season and they all have been away from home - to Cardiff, Aston Villa and Chelsea.
"The way we lost the game was incredible," added the Chilean, 60. "I think Chelsea, Cardiff and Aston Villa did not have chances during the games, but we conceded stupid goals."
Asked about the incident involving Nastasic and Hart, he added: "I do not talk about names, I talk about teams."