Specialists are due to remove the damaged glass and replace it with a new triple-glazed panel.
The council said it would work carefully to protect the archaeology underneath and preserve the scheduled ancient monument.
Repairs will cost nearly £29,830, most of which has been covered by an insurance claim.
Visitors will not be able to access the burial site during the work, but the rest of the visitor centre will be open as normal, the council said.
Of the accidental damage, a council spokesperson said: “A visitor sitting next to the grave site stood up, and the glass bottle of Perrier he had in the thigh pocket of his cargo trousers fell out and shattered on the glazed floor.
“The impact caused a small chip in the top layer of the triple glazed glass panel, which radiated out as a distinct crack across the panel.”
Assistant city mayor Vi Dempster apologised for the inconvenience, but said: “These essential works will ensure that the historic grave site of King Richard III continues to be protected to the very highest standards, while allowing visitors to experience this globally important heritage site.”