The man who died is thought to be in his early 50s.
The other man, reportedly in his 20s, suffered spinal and leg injuries and was taken to Addenbrooke's Hospital in Cambridge.
An investigation into the accident is underway.
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Any mist or fog will clear during the morning to give another dry day with light winds and some sunny spells.
The bus was travelling along Hale Street when it collided with a number of stationary cars and a lamppost before crashing into Sainsburys.
Rescue teams dashed to Trinity Street in Coventry shortly after 6pm after the Stagecoach bus crashed into the front of the store.