"Robertson, I would shoot for not earlier than mid-June," Cashman said Thursday.
Robertson currently is working out at the Yankees' training complex in Tampa. Cashman said Robertson still needs to do bullpens, batting practices and then rehab games.
"None of that has happened yet," said Cashman, who typically is cautious with injuries.
"He didn't think this was a serious issue and then the next thing you know, it turned into a DL and it has turned into something that is longer than 15 days."
Cashman said Robertson hasn't had any setbacks in the last couple of days.
"They are all guessing games," Cashman said. "If it is sooner, great. If it is later, then it is later.
"If you are trying to iron me down right now, I would say mid-June. He hasn't had any setbacks in the last 48 hours."
Robertson, who has been out since May 11, initially was given the opportunity to replace injured closer Mariano Rivera, who suffered a season-ending knee injury on May 4. Robertson went 1-for-2 in save opportunities before injuring his rib cage.
Robertson's extended stay on the disabled list has opened the door for Rafael Soriano, who has converted all six save opportunities this season and finished with an American League-leading 45 saves for Tampa Bay in 2010.
Both Cashman and manager Joe Girardi have raised the possibility that Soriano could keep the closer job when Robertson returns. Robertson would return to an eighth-inning setup job, a role he thrived in during his 2011 All-Star campaign.