The International Paralympic Committee (IPC) suspended Russia from the Rio Games starting on 7 September because of state-sponsored doping.
Cas then rejected a Russian Paralympic Committee (RPC) appeal against the ban earlier this week.
At least 266 Russian competitors are expected to now lodge individual cases.
The RPC's appeal, launched in the Swiss Federal Court, will be held on Monday.
Only Russian track and field athletes were banned from the Olympic Games, with individual sports' governing bodies deciding if they could compete, but the Paralympics ban includes all of the country's competitors.
Russian President Vladimir Putin said on Thursday the IPC's ban was "outside the bounds of law, morality and humanity".
Russia have contested the Court of Arbitration for Sport's (Cas) decision to uphold the country's blanket ban from the Paralympics.