It has been signed by the former world marathon champion and a number of other athletes, agents and promoters.
They believe the money belongs to sportsmen and women "whose earnings were reduced by cheating athletes".
Russia is suspended from international athletics because of widespread cheating and corruption.
It hopes to get the ban lifted in time for the 2016 Olympics, which begin in Rio de Janeiro on 5 August.
The online petition, which went live on Tuesday, had attracted 149 signatures by 14:00 GMT on Thursday.
Among them are marathon world record holder Radcliffe, Olympic heptathlon bronze medallist Kelly Sotherton, London Marathon chief executive Nick Bitel and Berlin Marathon race director Mark Milde.
Kenya's Edna Kiplagat, the 2011 and 2013 world marathon champion and Germany's Irina Mikitenko, who won the London Marathon in 2008, have also signed it.
"It's an extremely good idea," Bitel told the BBC's World Service. "There are many, many athletes around the world who've been affected.
"To try to leave it up to individual athletes and individual events to sue the Russian athletes who doped, that is just not practical."
