The duo want the Court of Arbitration for Sport (CAS) to overturn the decision of swimming's world governing body FINA to ban them for not falling within the IOC's new criteria on allowing Russian competitors to compete.
The criteria states that only if they have not previously failed a drugs test and can prove they are clean and not associated with the country's doping regime are Russian athletes eligible to take part in the Games.
Lobintsev and U.S.-based Morozov were part of Russia's bronze-medal winning 4x100m freestyle team at the London 2012 Games, while Lobintsev also won a silver medal in the 4x200m freestyle relay at Beijing 2008.
The two swimmers were suspended after they were named in relation to the "disappearing positives" revelations - false reporting of positive samples - in a World Anti-Doping Agency (WADA)-led independent report into doping in Russia.
Neither swimmer has ever served a ban for a positive test, and both have repeatedly said they are clean athletes.
Their bans came after the International Olympic Committee (IOC) last week handed the responsibility of banning Russian athletes back to the relevant sports federations.
Any Russian who has served a doping suspension is automatically ruled out but others were also to be banned if they could not effectively "prove themselves clean".
A statement issued by CAS on Saturday said the two swimmers had asked sport's highest court to declare the IOC ruling "invalid and unenforceable."
They also requested "that the IOC validates the entries submitted by the Russian Olympic Committee" - thereby reinstating them into the Russian team for the Games.
Lobintsev tested positive for meldonium earlier this year but was then cleared and given a "no fault" finding.
That was because of the decision by WADA that there was not enough information about how long meldonium takes to leave the body, and that athletes using it before it became illegal at the start of the year could not be punished.
Former world junior champion Daria Ustinova, also banned by FINA and who was given a warning over steroid use when she was 14, and four other swimmers who have previously served doping suspensions have not appealed.
