The campaign allegedly involved recruiting prominent figures to criticise the bids in their own countries, thus giving the impression they lacked support at home.
The newspaper says it was passed emails by a whistleblower who worked with the Qatar bid team.
Fifa rules say that bidders must "refrain from making any written or oral statements of any kind, whether adverse or otherwise, about the bids or candidatures of any other member association which has expressed an interest in hosting and staging the competitions".
Qatar's Supreme Committee for Delivery and Legacy said it "rejected" all the claims made by the paper.
According to the newspaper, the alleged smear campaign included paying a professor $9,000 (£6,900) to write a damning report on the economic cost of a World Cup in the US, recruiting journalists and bloggers to promote negative stories in American, Australian and international media, and organising grassroots protests at rugby matches in Australia.
The leaked documents also revealed that a group of American PE teachers had been recruited to ask congressmen to oppose a World Cup on the grounds the money would be better spent on high school sports, the paper claimed.
Lord Triesman, former chairman of the Football Association and England bid chairman, urged Fifa to "look at the evidence thoroughly", and said Qatar should not be allowed to "hold on to the World Cup" if they were shown to have broken Fifa rules.
He told the paper: "I think it would not be wrong for Fifa to reconsider England in those circumstances... We have the capabilities."
The Qatar bid team has previously been accused of corruption, but was cleared following a two-year inquiry by the Fifa ethics committee.
In a statement, Qatar's Supreme Committee for Delivery and Legacy said: "The Supreme Committee rejects each and every allegation put forward by the Sunday Times.
"We have been thoroughly investigated and have been forthcoming with all information related to our bid, including the official investigation led by US attorney Michael Garcia.
"We have strictly adhered to all Fifa's rules and regulations for the 2018/2022 World Cup bidding process."
Fifa has been contacted for comment.
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