Rummenigge blames the Russian billionaire’s spending on players and sky-high television rights deals for opening up a gap, even among Europe’s top clubs.
In a separate attack, he opened up a split in the ranks of the G14 group of elite clubs by branding the English, Italian and Spanish members as “selfish”.
Rummenigge, whose side lost Michael Ballack to Chelsea this year, told Kicker magazine: “I doubt whether in the next 10 years there will be a German club in the final of a European competition.
“Given what Abramovich does every summer in the transfer market, how can German clubs stay competitive?
“If Bremen get past Barcelona in the Champions League, that would be the eighth wonder of the world.
“Bremen earn €23 million from television rights and Barcelona get €143m.
“I just hope the European Union puts a stop to these excesses of Abramovich and foreign television.”
Rummenigge said Bayern, one of the founder members of G14 — which includes Arsenal, Liverpool and Manchester United — are now contemplating pulling out of the group.
He added: “Everybody in the G14 thinks only about themselves.
“I will say it absolutely clearly — (Ramon) Calderon (Real Madrid president) and his colleagues only fight for themselves.
“So do the Italians and the English. In football it is pure selfishness that rules. There is little common ground among the clubs.
“The way it is at the moment, we would have to think if it is still worth staying in G14.
“I am very unsatisfied with the way it is developing. There is no communication with FIFA, none with UEFA and there are too few common projects.”