Responding to Rodriguez's tweet, Cashman said to ESPN New York, "You know what, when the Yankees want to announce something, [we will]," adding, "Alex should just shut ... up. That's it. I'm going to call Alex now."
Rodriguez tweeted that he had been visited over the weekend in Tampa, Fla., by Dr. Bryan Kelly, who performed his January left hip procedure. Rodriguez said that Kelly "gave me the best news - the green light to play games again!"
But the Yankees have made no such announcement, and the team's medical staff -- not Kelly -- would have the final say as to when and where Rodriguez begins playing in Minor League games.
Rodriguez's tweet was accompanied by an Instagram photo of Rodriguez and Kelly sitting on a dugout bench at the Yankees' Himes Avenue Minor League complex. It is expected that Rodriguez will begin playing for Class A Tampa in the Florida State League sometime in July.
"I will continue to work hard until my efforts get me back in pinstripes and help my team win," A-Rod said in a statement in response to Cashman's remarks. "The tweet was pure excitement about Dr. Kelly's prognosis."
Cashman departed Yankee Stadium without speaking to other reporters. His comments to ESPN came after the GM was also irked by a newspaper report that cited a source close to Rodriguez saying he would begin to play in Minor League games on July 1. Cashman told MLB.com on Monday that was inaccurate.
"The tail is not wagging the dog, at least in this case," Cashman said.
The Yankees have provided only a vague timeline for Rodriguez's return, with the club sticking to its original expectation that he could be back in the lineup after the All-Star break.
Yankees manager Joe Girardi said that Rodriguez is making progress but will continue playing in simulated games until further notice.
"I don't tweet and I don't really follow Twitter, so I probably don't really know much of what's going on," Girardi said after the Yankees' 4-3 win. "As far as I know, he's not cleared [to play].
"There's always a chain of command with injuries and there has to be. That's the process. It goes to our training staff, our doctors, our GM and then it probably gets to me, so I'm kind of down on the totem pole."
Girardi also said that he does not expect the Yankees to receive any reinforcements from Tampa before the All-Star break, a rehabbing group that includes Rodriguez, shortstops Eduardo Nunez and Derek Jeter, as well as catcher Francisco Cervelli.
"This list is going on and on," Girardi said. "Guys are doing more every day."