The charity has written to Foreign Office Minister Bill Rammell to insist that inspectors from the committee that upholds Europe's anti-torture convention are allowed to make spot-checks on the prisons.
But both the UK and the Netherlands, which between them have thousands of troops in Iraq, have declared that the European Convention on the Prevention of Torture does not apply outside their own territories. They are the only two nations out of 45 that are signed up to the convention to have declared this limitation to it.
British authorities are investigating six deaths and 69 cases of injury and ill treatment by UK forces.
Director of public affairs at the foundation, Sherman Carroll, said: "Given the concern over the alleged ill-treatment of those detained in Iraq, we urge the Government to reconsider this restrictive declaration, and grant access to these detention centres. It is essential that steps are taken to safeguard the rights of detainees in Iraq."
The foundation said that the working arrangements of the Red Cross "do not allow for the transparency provided by the convention and its investigating committee, as it does not make its findings public, while the convention's investigating committee does."
Carroll added: "A secondary purpose of the visits would be to reassure public opinion worldwide that the UK Government gives full effect to its commitments under international human-rights law to protect detainees under its jurisdiction against torture."
It was a leaked ICRC report that sparked the furore over abuse by the occupation forces. Red Cross reports are normally shown only to the detaining authority.
An ICRC spokesman accepted that there had been criticisms of the organisation's policy of confidentiality, but added: "Our role is not helped by making public statements, which could mean that we would be deprived of contact with detainees."
The Medical Foundation treated more than 3,500 victims of torture in Iraq under the regime of Saddam Hussein.