Judge-led investigation would examine claims of British complicity in abuse and rendition of terrorism suspects.
"The inquiry is expected to offer compensation in cases, where necessary, and is likely to be held in private. A judge-led inquiry or commission may have the advantage of bringing together the 13 separate compensation cases currently going through the courts.
Those cases are leading to complex demands for the disclosure of documents that the intelligence services may not welcome, and are finding difficult to control. Some of the litigants have demanded an inquiry as part of their civil claims.
Cameron is understood to have discussed the issue in recent days with President Obama, but no decision is expected very shortly.
Tyrie said the inquiry needed to examine not only complicity in torture, but also involvement in the US rendition programme, while others said the inquiry should also embrace abuses perpetrated by British armed forced in Iraq."