Laurence Blairon, the ICC spokeswoman, told AFP: "The court counts on the co-operation of states and therefore of Qatar, but it does not have its own police force."
She said that Qatar may not be a member of the Rome Statute but that it was a member of the United Nations.
"The [UN] Security Council resolution that requires all states to co-operate with the court therefore applies to Qatar," she said.
Because the ICC cannot try al-Bashir in absentia, it depends on countries to execute their orders.
However, several African and Arab states have already called for the warrant to be suspended.
The ICC charged al-Bashir on March 4 with seven counts of war crimes and crimes against humanity on for his involvement in the Darfur conflict, which the UN estimates has killed between 200,000 and 400,000 people.
The Sudanese government puts the death toll at 10,000.
Al-Bashir is the first sitting head of state to ever receive an arrest warrant from the ICC.