Giant Eagle is tentatively scheduled to start site work on its on Pearl Road in mid-October, which puts the supermarket on track to open next summer.
Economic Development Director Brent Painter said the company is hoping to start construction on the building in early 2013.
Strongsville Golf said on its Facebook page it would close its driving range in October, but did not specify the date.
And Honey Hut Ice Cream, which is also on the site of the Market District store, has already built a new store in Brunswick to replace this one.
Painter said the company expects to start demolishing the structures and grading the site this month, and wants to have the pad ready for the the Market District building in February.
Building construction takes six to nine months, he said.
Market District stores are upscale supermarkets that feature a pharmacy, restaurant, a wide array of prepared foods and specialty departments including cheese, bakery and meats.
Voters agreed in March to rezone a portion of the Strongsville Golf property for the store.
The company went before the city's Architectural Review Board four times before getting design approval in July. The Planning Commission approved the plans a few days later.
Phil Bishop of Echo Realty said at that time that the company hoped to break ground in September or October.
The new store will open sometime during the summer montns, depending on how construction proceeds during the winter, he said.
"The main focus is to get started this fall and be open next summer," Bishop said.