A Wada-commissioned report found urine samples of Russian competitors were manipulated across the "vast majority" of summer and winter Olympic sports from late 2011 to August 2015.
The International Olympic Committee (IOC) will decide on Tuesday whether to provisionally ban Russia from the Rio Olympics, which start on 5 August.
IOC president Thomas Bach will host a telephone conference call to decide on "provisional measures and sanctions".
He has promised the "toughest sanctions available" and said the findings are a "shocking and unprecedented attack on the integrity of sport and on the Olympic Games".
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Dr Richard McLaren's report found 580 positive tests were covered up across 30 different sports, including in the build-up to London 2012, and at the 2014 Winter Olympics in the Russian city of Sochi.
McLaren said he had "only skimmed the surface" in his 57-day investigation, and Wada wants the Canadian to "complete his mandate" by identifying athletes who benefited from the programme.
The IOC should "decline entries, for Rio 2016, of all athletes" submitted by the Russian Olympic Committee and the Russian Paralympic Committee, says Wada.
Its president, Sir Craig Reedie, described the "scope and scale" of the findings as a "real horror story".
He told BBC Radio 5 live: "There are lots of potential samples to examine and that work will continue. We will almost certainly extend McLaren's mandate for him to do that."
Russia's track and field athletes are already barred from competing in Rio after the International Association of Athletics Federations voted in June to maintain a global competition ban on the All-Russia Athletic Federation (Araf).
Araf hopes to overturn the suspension and will find out by Thursday if its appeal to the Court of Arbitration for Sport has been successful.