Coach Dave Tippett had high hopes for an early return after watching Smith practice on Wednesday.
"I'd be surprised if he's not ready by the weekend," Tippett said.
Smith is eligible to be activated from injured reserve on Friday. Phoenix has a home-and-home series with Dallas on Friday and Saturday nights.
Smith left in the first period of last Wednesday's game against Columbus; Jason LaBarbera came on and led the Coyotes to a 5-1 win. The Coyotes lost 5-3 at San Jose on Thursday and 4-2 at home to Los Angeles two nights later with LaBarbera in goal; rookie Chad Johnson played Monday and shut out Nashville 4-0, earning the start in Wednesday night's home game against Edmonton.