The report says such a vote-swapping deal would have been in "violation of the anti-collusion rules".
The report says Cameron met FIFA vice-president Mong-Joon Chung (South Korea) in Prince William's suite at a Zurich hotel the night before the vote for the 2018 hosting rights in December 2010.
"The Prime Minister asked Mr Chung to vote for England's bid, and Mr Chung responded that he would if Mr (Geoff) Thompson (chairman of England's bid) voted for Korea (to host the 2022 tournament)," the report claims.
England officials in charge of organising the country's bid arranged jobs for the "adopted son" of FIFA vice-president Jack Warner at Aston Villa and Tottenham Hotspur (football clubs), and they were reportedly asked to engineer a meeting with the Queen for one FIFA official from South Africa. This official also raised the possibility of an honorary knighthood.
The report slams England's attempt to court Warner, who was also president of North, Central American and Caribbean football, alleging officials considered twinning his hometown in Trinidad and Tobago with an English village.
In 2009 the English Football Association also covered the costs of the Trinidad and Tobago Under-20 team when it stayed in Sheffield.
"England 2018's response shows an unfortunate willingness, time and again, to meet that expectation (of Mr Warner)," the report says.
England's hopes of hosting the 2018 World Cup were ended swiftly when its bid received only two votes, knocking it out of contention in the first round.
The report states there was "conduct by England 2018 that may not have met the standards set out in the FCE (FIFA code of ethics) or the bid rules.
"In many cases England 2018 accommodated or at least attempted to satisfy, the improper requests made by these Executive Committee members.
"While the bidding process itself, and the attitude of entitlement and expectation demonstrated by certain Executive Committee members in the exchanges discussed in detail above, place the bid team in a difficult position that fact does not excuse all of the conduct."
The first set of revelations from the so-called Garcia report painted a bleak picture of the background to the infamous 2010 vote that gave the 2018 World Cup to Russia and the 2022 World Cup to Qatar.
Garcia had resigned as head of FIFA's investigatory body in December 2014 in protest after FIFA released a 40-page sanitised summary of his report which he disowned, describing it as "incomplete and erroneous". The full report referred to an array of suspect financial dealings including the sum of $2 million allegedly sent by a consultant for Qatar, Sandro Rosell, to the 10-year-old daughter of a FIFA official.
Garcia's investigation also revealed that one former FIFA executive committee member thanked Qatar by mail for a transfer of several hundred thousand euros just after Qatar was awarded the 2022 tournament.
The report also documents that three executive members of FIFA were flown to Rio de Janeiro for a private party ahead of the vote to decide who would host the 2022 World Cup.
