"The vast majority of that material has been available to us for some time," Garcia told FIFA's congress of 209 member countries on Wednesday.
The Sunday Times' reports in the past two weeks revived calls to re-run the December 2010 vote by FIFA's executive committee. The newspaper promises further revelations during the World Cup.
On Monday, Garcia wrapped up his lengthy probe of the 2018 and 2022 World Cup hosting contests.
The deadline prompted reports that Garcia and his team were prepared to ignore further Sunday Times revelations.
"We have gone to the original source, what appears to be the original source, and we are confident we will have access to that," the former U.S. Attorney said.
Judging by further Garcia's comments Wednesday, German football icon Franz Beckenbauer — a FIFA voter in 2010 — could face disciplinary action following reports that he has refused to co-operate with the case.
Garcia is due to deliver his report next month to FIFA ethics judge Joachim Eckert, who can recommend sanctions.
Allegations implicating FIFA board members in bribery, voting pacts and favour-seeking circled even before the vote. Qatar beat the United States 14-8 in a final round of voting.
The Sunday Times reported it received a massive cache of Qatari official bin Hammam's emails and documents leaked by "a senior FIFA insider."
The newspaper alleged bin Hammam paid $5 million to African football leaders for supporting Qatar's bid and his own FIFA presidential ambitions.
It also claimed bin Hammam set up government talks for a natural gas deal potentially worth tens of millions of dollars for Thailand, home country of his longtime FIFA board ally Worawi Makudi.
Bin Hammam survived a FIFA election bribery scandal when challenging Blatter in 2011, before being expelled the next year for financial wrongdoing while leading the Asian Football Confederation. A forensic audit by PricewaterhouseCoopers revealed extensive details of bin Hammam's deals.
Beckenbauer was named in The Sunday Times for meeting with bin Hammam in Qatar and taking consultancies with German industry seeking contracts linked to World Cup projects after he took part in the FIFA vote.
Garcia said all football officials are required by FIFA's code of ethics to co-operate with his investigation team.
"And it makes real penalties available to all those who fail," he said during an eight-minute speech.
Beckenbauer was reported saying at a news conference in Munich last week that he twice declined Garcia's offers to meet. He is an advisor to FIFA's football committee, and still represents Bayern Munich around the world.
Garcia, a former Interpol vice-president, was appointed by FIFA in July 2012 to investigate allegations of wrongdoing.
He was urged by FIFA's anti-corruption advisors to make the 2018 and 2022 World Cup hosting contests a priority. FIFA gave him an unlimited budget and time to investigate cases.
Garcia said his report would be "comprehensive and fair to all parties."
Since last October, Garcia and his team have interviewed officials from the nine bid candidates, and reviewing evidence from whistleblowers.
Qatari officials were the last to be visited, in Oman 24 hours after the first round of Sunday Times allegations.
Garcia delegated his Swiss deputy, Cornel Borbely, to quiz officials from his home country and Russia. There, Garcia is barred from entry for his previous work prosecuting a Russian arms dealer.
Garcia said he also spoke, or attempted to speak, with all members of the FIFA executive committee from 2010. Several have since resigned or been suspended while under investigation for financial wrongdoing.
After Garcia arrived in Zurich to meet with current board members in March, some reportedly agitated to remove the American from his post.
"We did not swoop in for surprise interviews," Garcia said.