In the Lebanese capital, the remains of the 32-year-old orthopaedic surgeon would be handed over to the British embassy.
"The body will be handed over to the ICRC at a Damascus hospital and then taken to Beirut," spokesman Simon Schorno told AFP early today.
Khan's body was due at the frontier with Lebanon at midday (1530 IST), where it would be transferred to another ICRC team.
Previously, Syrian Deputy Foreign Minister Faisal Muqdad told AFP the body would be transported to Beirut by yesterday "at the latest".
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Muqdad had also said the ICRC attended a final autopsy on Khan's body, but Schorno said his organisation "did not take part in any autopsy."
London has held Damascus responsible for Khan's death.
But Syrian authorities said on Wednesday the doctor was found "hanging" in his cell, where he was being held for "unauthorised activities," and that he had committed suicide.
Muqdad said President Bashar al-Assad had decided to grant Khan an amnesty and hand him over to his mother and British MP George Galloway at a news conference in Damascus.
Khan, a volunteer with London-based charity Human Aid UK, had travelled to Aleppo in northern Syria last year to help civilians when he was arrested by the regime.