The appeal will be against the decision of the International Association of Athletics Federations (IAAF) to ban them from this year's Olympic Games in Rio de Janeiro.
ARAF secretary general Mikhail Butov revealed they are currently putting together a legal team to lead the appeal.
Among those who may work on the case is London-based law firm Morgan Sports Law.
The firm recently successfully led an appeal to CAS from Tatyana Andrianova against the IAAF and ARAF against a decision to strip her of the bronze medal she won in the 800 metres at the 2005 World Championships following a re-analysis of her urine sample which had shown traces of banned performance-enhancing drugs.
They also successfully appealed to CAS on behalf of Belarus' Olympic hammer silver medallist Vadim Devyatovskiy to have a lifetime ban imposed by the IAAF from the sport lifted, despite him having been involved in several doping scandals during his career.
Russia's two-time Olympic pole vault gold medallist and world record holder Yelena Isinbayeva is among the athletes expected to appeal individually to CAS.
"A suit or suits will be submitted to CAS next week," Butov told Russia's state news agency TASS.
"It is also necessary to completely determine the legal direction, but we have generally formed a group of lawyers with whom we will work.
"I don’t known whether Mike Morgan of the UK will be among them, so far I’d rather not disclose the names.
"Class actions will be filed for certain, but some individual suits are also possible.
"We are going to have consultations with these lawyers in the coming days.
"Then we will make the final decision."
The IAAF ruled last Friday (June 17) that a ban imposed on the ARAF after the publication of the World Anti-Doping Agency (WADA) last November following allegations of state-supported doping should not be lifted in time for this year's Olympic Games in Rio de Janeiro.
It was decided that Russian athletes who had operated outside the country and could demonstrate they had gone through enough anti-doping controls should be allowed to compete under a neutral flag at Rio 2016.
The International Olympic Committee (IOC), however, have claimed there is no rule that allows for that to happen and they must compete under the Russian flag.
Sir Craig Reedie, the President of WADA, has backed the stance of the IAAF, even though he is also vice-president of the IOC.