On Sunday at Atlanta, Cozart was hit on the wrist by a Julio Teheran fastball in the first inning and a ground ball struck his finger in the fourth inning which knocked him out of the game.
Kristopher Negron started a third straight game at shortstop for Cozart in Thursday's 7-2 loss to the Pirates.
Cozart passed a test Wednesday when he was able to make 40 throws without problems. On Thursday, he started swinging the bat.
"There is still inflammation in there right now, but it just feels way better," Cozart said of his wrist. "I swung, took soft toss, and did my normal routine and it felt really good."
Cozart took batting practice with the team before the series finale vs. the Pirates. He also hoped to test his wrist in the field by backhanding ground balls. If there are no issues, he could be back for the next series that opens on Friday vs. the White Sox at U.S. Cellular Field.
"I think it's more realistic to look and see where he is [Friday] as far as a potential start," Price said.
Mark Sheldon is a reporter for MLB.com. Read his blog, Mark My Word, and follow him on Twitter @m_sheldon. This story was not subject to the approval of Major League Baseball or its clubs.