The arrests were made at 0545 BST at addresses in London after counter-terrorism officers received intelligence of a potential threat.
The five men have been taken to a central London police station.
Officers are continuing searches at premises connected to the raids. The men are not British nationals.
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The five were arrested on suspicion of the commission, preparation or instigation of acts of terrorism under the Terrorism Act 2000.
The five men are 26, 27, 36, 40 and 50 years old. They were arrested in an armed operation at business premises where searches are continuing.
Residential premises in north and east London are also being searched. Officers have not found any hazardous items.
It is not clear whether the investigations relate to a plot against the Pope himself or an element of the visit.
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