Citing members of Congress and their aides, The New York Times reported on Saturday that the requests were sent last week by House of Representatives and Senate Intelligence Committee members.
Goss has resisted publicising the report, in part because it could have a negative effect on agency morale, The Times reported.
Some dozen current and former intelligence officials, including former director George Tenet, are said to be singled out in the report.
CIA inspector general John Helgerson prepared the report at the request of the joint congressional committee that in 2002 completed its review of the attacks, The New York Times said.
Goss sent a classified version of the document to Capitol Hill in August, and Helgerson briefed the intelligence committee on the findings, the paper said, but distribution has otherwise been sharply limited.
A CIA spokeswoman declined to comment on whether Goss might release the report to the public, adding that Helgerson had complied with the law in preparing the report and delivering the findings to Congress.
Republican officials who told the Times that the report should be made public include Repulican Peter Hoekstra of Michigan, who chairs the House Intelligence Committee, and Senator Pat Roberts of Kansas, chairman of the Senate Intelligence