Armed officers were sent to a store in Blackheath, in the West Midlands, at about 12:10 BST and cordoned off the scene as a helicopter hovered above.
Officers searched the area and a man was detained. Police do not believe the incident is terror related.
The supermarket was evacuated and several roads closed with drivers urged to avoid the area.
No injuries have been reported, West Midlands police said.
Ch Insp Paul Minor said: "The store remains closed while we examine the scene and assess the cause of the fire... we do not believe this to be related to any terror threat."
Halesowen Street, John Street and Union Street were closed.
In a statement, Sainsbury's said: "We can confirm there was an incident at our Blackheath store, which is currently closed.
"Thankfully no colleagues or customers were harmed and the matter is now being investigated by West Midlands Police."
The UK terrorism threat level was reduced from critical to severe earlier, five days after the Manchester Arena attack on Monday, but increased security remains in place at events around the country.
A huge police operation is under way in Birmingham while the city's annual two-day Pride festival takes places.
A man carrying a knife was seen running from a Sainsbury's store after a fire was started in the toilet, police said.