Garcia will file a final report to the judging division of FIFA's ethics court to decide on possible sanctions.
He is expected in England this week to speak with members of its 2018 bid team, in the first of a series of visits to the 11 countries involved in nine separate bids.
"Members of the Investigatory Chamber intend to speak with and request information from representatives of every bid team that vied to host the 2018 or 2022 FIFA World Cup," Kirkland and Ellis said. "The fact that we request a meeting with members of a particular bid team does not mean that any specific allegation has been made by or against that team or anyone associated with it."
Leaders of the Qatar and Russia bids said last week in Zurich, on the sidelines of FIFA's ruling board meeting, they have not yet been contacted by Garcia's team.
Garcia's Swiss deputy, Cornel Borbely, will lead the American and Russian questioning "to avoid the appearance of any conflict of interest."
Qatar defeated the United States 14-8 in a final round of voting.
Russia barred Garcia, a former U.S. Attorney, in April in retaliation for his previous work prosecuting a Russian arms dealer, Viktor Bout.
Bout is serving a 25-year prison sentence in the U.S.
Garcia's statement on Tuesday reminded potential witnesses still serving in official football duty that they are obliged by the FIFA ethics code to cooperate with his requests for help.
Anonymity can also be offered to some witnesses "in appropriate circumstances," Garcia's statement said.
Garcia's office has been receiving information since February from a whistleblower hotline created by FIFA for officials and fans to allege ethics violations.
Russia's bid, which was strongly supported by state president Vladimir Putin, also beat joint candidacies from Spain-Portugal and Belgium-Netherlands in the 2018 contest. Australia, Japan and South Korea were the other 2022 candidates.