Construction of the new Market District store on Pearl Road is on schedule, according to the company.
"They expect to open toward the end of the year," said Matt Metzger of Adcom Group, which serves as a spokesman for Giant Eagle.
Voters in March 2012 cleared the way for Giant Eagle to on the former Strongsville Golf property by rezoning the property.
So far, crews have razed the buildings on the site -- including Strongsville Golf and Honey Hut -- and have cleared trees and graded the land.
Phil Bishop of Echo Development said in December that his company, which is getting the land ready for construction, plans to deliver the parcel, with a concrete pad that will house the building, to Giant Eagle in March.
Strongsville Community Development Director Brent Painter said he has spoken with project officials and found they are pleased with the pace of the work.
"They said things are going well," Painter said. "Everything's on schedule."
In addition to the grocery store, Giant Eagle is also working on plans to build a Get Go gas station at the corner of Drake and Pearl, on the site of a former service station.