Romo suffered his latest injury in the third quarter of Sunday's 20-10 victory over the Philadelphia Eagles.
He is expected to miss eight to 10 weeks based on preliminary indications, according to a source.
Romo will undergo more tests Monday, including a CT scan, a Cowboys official said. The tests will determine whether he needs surgery.
"We have to determine how much muscle damage there is in the shoulder," Cowboys coach Jason Garrett told ESPN's Sal Paolantonio. "Then we will have a better idea on the length of time he will miss."
Romo was hurt with 9:44 left in the third quarter after a sack by Eagles linebacker Jordan Hicks.
"I felt it," Romo said, his arm in a sling. "It had a little bit of a pop to it. Obviously, I've felt it before. Your body goes into a little bit of shock and you just have to wait it out for a second to come back."
Garrett did not offer a possible timetable for Romo's return. At eight weeks, the quarterback could be back Nov. 15 against the Tampa Bay Buccaneers. At 10 weeks, he could return on Thanksgiving against the Carolina Panthers or Dec. 7 against the Washington Redskins.
It is possible the Cowboys could place him on the injured reserve/designated to return list, which by rule would keep him out for eight weeks.
Romo will join star receiver Dez Bryant on the Cowboys' sideline. Bryant had surgery last week after breaking his right foot in the season-opening win against the New York Giants. He is expected to be out four to six weeks but could need longer to recover.
"We said a lot of prayers while they were evaluating [Romo]," Dallas owner and general manager Jerry Jones said. "We never would've thought after two ballgames we'd have no Tony Romo, no Dez Bryant, even no [Randy] Gregory, the way he was playing. On the other hand, this is the life we chose and we know good and well this can happen to you."
Hicks landed on top of Romo, forcing the quarterback's shoulder into the ground. Romo then remained on the ground for a few moments before heading straight to the locker room for an X-ray.
"It just usually times up perfectly with body weight hitting you in a certain angle," Romo said.
In 2010, Romo suffered the same injury when he was driven into the turf by Giants linebacker Michael Boley in Week 7. Romo did not have surgery to repair the collarbone. The team held out hope Romo could return later that season, but he was eventually placed on injured reserve Dec. 22 with two games to go in a lost 6-10 campaign.
Brandon Weeden will replace Romo as the starter. He completed 7 of 7 passes for 73 yards Sunday, including a 42-yard touchdown to Terrance Williams with 4:13 to play. The Cowboys lost Weeden's only start last year to the Arizona Cardinals after Romo suffered two transverse process fractures.
"We are going to run our offense," Garrett said. "Obviously Tony's a special player and a great leader for our football team for a number of years now. But guys have to go in there and play. Brandon did a good job today. He has done a good job in the past when he has opportunities to run our offense. Our standards are high for anybody who goes in and plays. He understands that. He will prepare well and play well."
Las Vegas sportsbooks reacted quickly to Romo's injury even before the Cowboys completed their victory.
Before the injury, the Westgate SuperBook listed the Cowboys as 4.5-point favorites over the Atlanta Falcons next week. After the injury, the book adjusted the line to a pick 'em.
Romo's injury also prompted the MGM sportsbook to move the Cowboys from 8-1 to 10-1 to win the Super Bowl.
"They've got two division wins in their pocket," MGM vice president of race and sports Jay Rood told ESPN.com's David Purdum. "That's big."
Also Sunday, the Cowboys said Jason Witten has two sprained ankles and a sprained knee.
"It was one of those nasty games -- I have some bumps and bruises and rolled the ankle a little bit -- I'll be fine," Witten said. "Just typical football. Sometimes those things happen."
ESPN.com's David Purdum contributed to this report.
Romo: I'll be back
Dallas Cowboys QB Tony Romo and coach Jason Garrett discuss the quarterback's broken clavicle in Sunday's 20-10 win over the Eagles.
Romo suffers broken collarbone in Cowboys' win
Tony Romo breaks his left collarbone in the third quarter and would have to exit the game as the Cowboys top the Eagles 20-10.