The Court of Arbitration for Sport (CAS) will hold an emergency session in Rio on Sunday to hear there case, just five days before the start of the Games, according to sources close to the case.
The move by the two Olympic medal winners was announced as the International Olympic Committee held two days of talks on fallout from the Russia doping crisis.
Morozov, 24, and Lobintsev, 27, have called on CAS to declare "invalid and unenforceable" an IOC order for federations to exclude athletes implicated in an investigation on Russia's state-run doping system. They were among seven Russians banned by the International Swimming Federation (FINA) last week after the order.
A report by Canadian lawyer Richard McLaren for the World Anti-Doping Agency (WADA) said Russia's doping had been organised by the sports ministry and aided by the Russian secret service at the 2014 Sochi Winter Olympics.
IOC panel to decide Russia team
So far, at least 117 individuals from the 387 that the Russian Olympic Committee wanted to enter have been excluded. Russian sports minister Vitaly Mutko said Saturday he expected 266 athletes to compete. Boxing, golf, gymnastics and taekwondo federations have yet to report their decisions.
A three-member IOC panel made up of Ugur Erdener, president of World Archery and head of the IOC medical and scientific commission, Claudia Bockel of the IOC athletes commission, and Spanish IOC member Juan Antonio Samaranch, will make the "final decision" on the Russian team, IOC spokesman Mark Adams said.
Morozov, a member of the 4x100m freestyle relay team that took bronze at the 2012 London Games, and Lobintsev, who took silver in the 4x200m freestyle team in Beijing in 2008 and bronze in the 4x100m freestyle in London, have taken their action against the IOC and FINA.
"Both swimmers request CAS to declare the decision of the IOC executive board of July 24 2016 invalid and unenforceable," said a CAS statement.
"The swimmers also request that the decision of the FINA bureau of July 25 2016, declaring both of them ineligible for the Olympic Games in Rio, be set aside."
Morozov said in a letter to FINA president Julio Maglione this week that he had never failed a drug test taken by Russian and international experts.
"Throughout the last six years I've been drug tested by doping control agencies at my home and at the pool, at least once a month, and sometimes every other day," he said in the letter published on his Facebook page.
"I am sure that in a justice-driven system I have full right to take part in the Olympic Games."
WADA president Craig Reedie, who called for a complete ban on Russian athletes in Rio, is to address the IOC meeting on Sunday. The CAS has already rejected an appeal made by 67 Russian athletes against a ban ordered by the International Association of Athletics Federations (IAAF) before the IOC sanctions.
Athletics was the first sport touched by the doping controversy. Russian doping whistleblower Vitaly Stepanov told a Brazilian newspaper that the Rio Olympics "will not be clean" and blasted the IOC for not banning Russia.
Stepanov, who with his 800m runner wife Yuliya Stepanova, gave details of the state-run doping programme to a German documentary released in 2014, said efforts to clean up sport had failed.
"It has always been the case in the Olympics. There has never been a clean Olympics and there is no reason to believe that Rio will be clean," he told O Estado de Sao Paulo newspaper.
"Unfortunately, doped athletes will be competing," said the former Russian anti-doping agency (RUSADA) official now living in hiding in the United States with his wife.