Yankees manager Joe Girardi said he thought Thursday was a more likely scenario, but Tampa-based team physician Dr. Allen Miller could tip the outlook in Rodriguez's favor.
"If [Miller] thinks it's realistic, then that's what we'll go for," Girardi said. "The sooner we get him, the better off we are."
The Yankees had hoped Rodriguez would be able to play in Minor League rehab games this week, but an MRI exam taken on Monday showed a strain -- a reduction and improvement, but still a strain.
Accordingly, Rodriguez will miss this weekend's Subway Series against the Mets. He will remain behind in Tampa when the Yankees travel north following the final game of their four-game set with the Rays on Thursday.
"I was really looking forward to it. What better time to come back [than] at home for the Subway Series?" Rodriguez said. "Obviously, I'm disappointed, but I have to keep a level head about it and realize that there's so many more games that are so important. To play three to lose 60, the risk is not what we want."
Rodriguez sustained the injury running out a ground ball on April 20 at Baltimore. He aggravated the quadriceps when he returned to the lineup after missing three games, and has not played since April 28.
Rodriguez again took batting practice, fielded grounders and ran in the outfield on Tuesday at the Himes Avenue Minor League facility.
"He pushed it a little bit harder running and said he felt good," Girardi said. "He actually said he felt better than yesterday. He had a good day."
The Yankees' main concern is the first aggressive burst Rodriguez would make after jumping out of the batter's box at game speed, which was how he re-aggravated the injury against the Orioles. New York was 5-7 entering Tuesday's game without Rodriguez.
"You miss him. It's a huge bat against left-handers, against right-handers, all the time," Girardi said. "Those guys aren't replaceable, but you've got to find a way to get it done. Other guys have to pick up the slack."
In other injury updates, catcher Jorge Posada continued to throw in the outfield at Tropicana Field, with a June 1 return still realistic. Girardi also caught last year's first-round Draft pick, right-hander Andrew Brackman, in a bullpen session as the 6-foot-10 hurler comes back from Tommy John elbow ligament replacement surgery, and looked on as right-hander Humberto Sanchez threw a bullpen session with a June 10 return to competitive game action in sight.
"I thought it was important to get them over here," Girardi said. "It's long being on rehab."