Algeria announced on Monday that Gaddafi's wife, two of his sons and his daughter had crossed into its territory, prompting Libya's ruling interim council to demand that they be handed back to face trial.
Quoting Algerian sources, the el-Chorouk newspaper said on its website that President Abdelaziz Bouteflika had told government ministers during a cabinet meeting on Monday that Algeria would respect international law on all matters related to the Libyan conflict.
"Should Gaddafi try to enter Algerian soil amid talk that the rebels are tightening their grip on the border with Tunisia, and Egyptian (border) restrictions, Algeria would arrest him and hand him over to the International Criminal Court in compliance with international agreements," the newspaper said.
It said the decision was not a reaction to the toppling of the Gaddafi regime but was in accordance with the ICC's arrest warrants for Gaddafi, his son Saif al-Islam and his intelligence chief over accusations they committed crimes against humanity.
No one was immediately available for comment from the government.
In a separate report, el Chorouk said Gaddafi's family members, who had crossed into Algeria, were in the southeastern Illizi province, which borders Libya, and would not be allowed to go to the capital Algiers.
Gaddafi's location remains unknown a week after Tripoli fell to his foes.
Meanwhile, Algeria's El Watan newspaper said the country was closing the southern part of its border with Libya. Quoting diplomatic sources, El Watan said an instruction had gone out to security services to close the southern part of the border with Libya, because of the "precarious situation" there.
The Algerian government could not immediately be reached for comment. Algeria's border with Libya is hundreds of kilometres long and stretches through vast expanses of empty desert.
Algerian officials say they are concerned that Islamist militants have infiltrated Libya's National Transitional Council (NTC), and that al Qaeda's North African wing will exploit the chaos in Libya to acquire weapons and explosives.
The NTC has angrily denied those charges and in turn accuses Algeria of siding with Muammar Gaddafi during Libya's civil war, something the Algerian authorities deny.