The AMA suspended the RUSADA in 2015 after declaring that it does not comply with the revelations of a vast scheme backed by Moscow to avoid the drug evaluators.
A WADA report by Canadian lawyer Richard McLaren accused the Russian authorities of running an elaborate doping program with the full support of the Russian Ministry of Sports and the Russian secret service or FSB.
The Russian authorities had concluded that their doping system had failed, but refused to acknowledge the claims of institutional participation.
Tensions within the AMA emerged earlier this year about whether Russia had done enough to lift the sanction, and some officials argue that it had not completed a "road map" towards rehabilitation.
However, the AMA statement on Friday said that its compliance committee had recommended reinstating RUSADA after reviewing a letter from the Russian Ministry of Sports.
"This letter sufficiently recognized the problems identified in Russia, thus fulfilling the first of the two outstanding Roadmap to Compliance criteria of RUSADA," said WADA.
He said that Russia also agreed to provide access to data and samples in its Moscow laboratory to WADA through an independent expert.
But the apparent change of attitude of the AMA was criticized by the head of the Anti-Doping Agency of the United States, Travis Tygart, who asked the agency to make public the evidence that Russia had met the reincorporation criteria.
"Frankly, it sucks heaven, the AMA must stop the sleight of hand and launch the new recommendation of the Compliance Compliance Committee, as well as any information received from Russia that demonstrates that they meet the requirements," Tygart said in a statement.
"Today, the AMA has unequivocally told the world what kind of organization it is: one that supports the wishes of a handful of sports administrators about the rights of millions of clean athletes."
Tygart also challenged RUSADA to provide details of the tests conducted in his Moscow laboratory.
"If RUSADA complies, then it's great, now we have all the data and samples in the Moscow laboratory and, finally, justice can be done in the hundreds of cases that have been derailed so far," Tygart said. "Otherwise, the solution has been present from the beginning."
If the reinstatement of RUSADA is approved as expected next week, it could have far-reaching implications throughout the sports world.
The most significant thing is that it eliminates a key obstacle to lifting the suspension of Russia by the International Association of Athletics Federations (IAAF), world governing body of athletics.
The IAAF has taken a tough line on Russian athletes since the scandal broke out, refusing to lift its ban shortly before the European Championship.
The IAAF has said that Russia must meet three criteria to be readmitted; the restitution of RUSADA, provide access to doping records in the agency's laboratory in Moscow from 2011 to 2015, and the payment of costs incurred by the IAAF in the wake of the scandal.
The IAAF will once again review the status of Russia at its council meeting in Monaco in December.
On Thursday, a group of British athletes had asked WADA to resist pressure to lift the ban, claiming that Russia was not complying.
The Anti-Doping Commission of the United Kingdom said in a letter to the head of the AMA, Craig Reedie, that the readmission of Russia would be a "catastrophe for clean sport".
Clear divisions within the AMA came to light at its board meeting in Montreal in May, where members of individual sports bodies called for lifting the ban.
Patrick Baumann, a prominent member of the International Olympic Committee, questioned whether Russia should be suspended indefinitely.
"We do not challenge the road map, we just ask how long we want to follow that road map, for the next 10 years, 20 years, 30 years?" He told the meeting.
Meanwhile, officials in Russia have sent contradictory signals about the country's doping scandal.
Russian President Vladimir Putin has acknowledged the cases of doping but has fired Grigory Rodchenkov, the informant whose revelations exposed the scandal as "an idiot" who should not be trusted.
Rodchenkov currently lives in hiding in the United States and has said he fears for his life.