Chelsea supporters sang for the coach known as the 'Special One' during Saturday's 1-1 FA Cup draw with second-tier Birmingham City after the Inter faithful had done the same in Friday's 3-0 Serie A defeat at home to Bologna.
The results piled more pressure on Chelsea's Andre Villas-Boas and Inter's Claudio Ranieri who was also succeeded by Mourinho at Stamford Bridge back in 2004.
"My message to the fans is I love Chelsea and Inter a great deal but I would prefer them to support their own teams and coaches," Mourinho told a news conference after Real's 4-0 La Liga win at home to Racing Santander.
"I am the past, a very good past, but I am not there anymore and they should all be fighting together."
Mourinho led Chelsea to the English Premier League title in 2005 and 2006. His next stint as a coach at Inter then yielded an unprecedented Serie A, Italian Cup, Champions League treble in 2010.
Villas-Boas, who used to work under Mourinho at Chelsea and Inter, has come under fire with the London club fifth in the table having won just two of their last 10 league games.
Ranieri is in a similar situation with Inter sixth in Serie A after a run of three straight defeats.
