The IAAF, track and field's world governing body, on Friday upheld a suspension of the Russian team imposed in November after a World Anti-Doping Agency report detailed widespread, state-sponsored doping.
Race walkers Denis Nizhegorodov and Svetlana Vasilyeva argue that a ban of the entire team is unfair punishment.
Nizhegorodov, an Olympic silver medalist in 2004, says "competing at the Olympics is the main goal and main honor. We will get that right," in comments on agent Andrei Mitkov's website.
Both athletes say they reject an IAAF measure that would allow some Russian athletes to compete under a neutral status, rather than the Russian flag, if they can show they are clean and have been tested regularly by a reputable testing authority outside Russia.
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"I'm a citizen of Russia, a great sports power. I don't agree with competing under the Olympic flag," he said. The appeal does not have the blessing of the Russian track and field federation, which says it has yet to receive the full text of the IAAF's decision.
Nizhegorodov comes from a training center that has seen over 30 doping cases, including several for Olympic champions, and where senior officials have been banned for organizing drug use.
He was threatened with losing his Olympic bronze from 2008 after a retest came back positive, but was cleared when the "B'' sample did not correspond with the original finding.
The appeal by Nizhegorodov and Vasilyeva came as IAAF President Sebastian Coe defended the decision to ban Russia, saying it should not be seen as an attempt to stop clean Russians from competing.
Coe also provided further details on an IAAF program to allow a select few Russians to compete if they have been based outside the country with testing from a rigorous foreign anti-doping agency.
"Some of them made the tough personal decision to get out and are now training in systems that are effective and safe," Coe wrote.
The International Olympic Committee has convened a summit for Tuesday to discuss the issue but said in a statement Saturday that it supported the IAAF ruling.
Russian pole vault world record holder Yelena Isinbayeva said that next week's Russian national championship would be the last competition of her career if the ban is not lifted in time for the 34-year-old athlete to go to Rio.
"If IOC won't allow us to participate in the Olympic Games, it will be my final (competition) in the career," she wrote on Instagram. Isinbayeva is a two-time Olympic gold medalist and won bronze in 2012.