Drafted second overall in the 2012 MLB First-Year Player Draft, the talented outfielder has yet to progress to Double-A.
"We've said all he's got to do is put up some numbers, then we'll consider it," general manager Terry Ryan said. "He's going to get us to start thinking here pretty soon."
After returning from a left wrist injury on July 6, Buxton went 0-for-8 in his first three games with Fort Myers, but he has been flashing his elite talent of late.
Thursday's home run was his first since returning, and he had a triple during a 3-for-4 performance on Wednesday.
"I hope he keeps it up," Ryan said.
Alex M. Smith is an associate reporter for MLB.com. Follow him on Twitter @asmiff. This story was not subject to the approval of Major League Baseball or its clubs.