The Gators (2-0) made the announcement Thursday, two days before they open league play at Kentucky (2-0).
Grier and sophomore Treon Harris split snaps during the first two games, with each getting a start. Grier has completed 26 of 35 passes for 317 yards and four scores. He also has an interception and a fumble.
Harris has completed 19 of 27 for 269 yards and two touchdowns.
McElwain said Wednesday he doesn't have a timeline for picking a starter for the rest of the season and plans to go with the hot hand. McElwain says, ''If a guy's hitting it and distributing it and has a good stroke, then we'll go with it.''