The fire service says air is being pumped through Costa & Laura Ashley - on the other side of the building - to stop the fire spreading.
About 110 firefighters' are at the scene and a cordon has been set up.
A 70 tonne demolition machine called a high reach excavator arrives on Cathedral Green in Exeter to work at the Royal Clarence Hotel early on Tuesday morning. It was a slow and careful operation. Meanwhile an aerial fire platform was still pumping water into hotspots in the hotel.