Tesco wants to refurbish the former Boots store – a listed building – at the west end of the city thoroughfare.
More than 20 jobs will be created if the Tesco Express outlet is given the green light.
A spokeswoman for the chain said: “We are delighted to be investing in Edinburgh with a new Tesco Express store at the west end of Princes Street.
“We are pleased to be bringing back to life a vacant unit and adding to the vibrancy of this world-famous street.
“The Princes Street store will offer a fantastic range of quality products including fresh groceries, toiletries and household essentials.”
In November last year, Tesco was given the go-ahead to sell alcohol at the planned store, despite objections from NHS Lothian.