Raiders coach Jack Del Rio said Carr's injury "could be as short as two weeks -- it could be longer." Del Rio added that Carr could be out as many as six weeks.
"He said he's sorry," Del Rio said of Carr. "Great kid. He'll bounce back. I told him that the team will take care of business while he's healing and just get healed up and when he can come back, he'll come back."
Carr was injured on a sack late in the third quarter of Oakland's 16-10 loss at the Denver Broncos on Sunday and did not return to the game.
After being twisted down awkwardly by Adam Gotsis, Carr grabbed at his lower right back and stayed prone on the ground. After several minutes, he was helped up and walked slowly off the field. After being examined in the blue pop-up tent on the Raiders' sideline, he walked back to the Oakland locker room.
Transverse processes are small projections on the vertebrae where soft tissue attaches, but they have no real role in load-bearing. In football, fractures to transverse processes are not uncommon when there is a direct hit, like a helmet or knee to the back, resulting in bruising and pain at the fracture site.
"Pain," Carr said Sunday, describing what he felt when he went down. "My back didn't feel too good."
Carr suffered a fractured pinkie on his passing hand in late November last season before a broken right fibula ended his season in Week 16. He still finished tied for third in NFL MVP voting.
In June, Carr signed what was then the richest contract in NFL history -- a $125 million deal over five years -- and was off to an uneven start this season, with the Raiders losing two straight games, at Washington and Denver, in dispiriting fashion after two wins to open the season, at Tennessee and against the New York Jets.
Carr has completed 67.9 percent of his passes for 753 yards and seven touchdowns with two interceptions.
As of now, a team source told ESPN's Chris Mortensen the Raiders have no plans to add a quarterback to their 53-man roster as they monitor Carr's healing. EJ Manuel and Connor Cook will both be active while Carr is sidelined. The Raiders do not have a quarterback on their practice squad.
Carr's brother David, who works for the NFL Network, said Carr was surprised by the diagnosis.
"He's bummed out, as you can expect," David Carr said on the network. "He didn't see this coming. He thought it would be something that he could work through, get loosened up by Wednesday or Thursday, go out and practice."
Players can typically return to play as the pain allows, even as the bone continues to heal because of the low risk involved.
Panthers quarterback Cam Newton suffered two transverse process fractures in a car accident in 2014 and missed one game. Former Cowboys quarterback Tony Romo also missed only one game after suffering two transverse process fractures in a game in 2014, the result of a knee in the back by Washington linebacker Keenan Robinson.
With Carr out, Manuel will step in as the team's quarterback for Sunday's home game against the Baltimore Ravens.
"I think the thing that we have seen in watching EJ Manuel is he's running the same offense that Derek Carr was running," Ravens coach John Harbaugh said Monday. "So, it's not like they have a different offense for Manuel. He's running the Oakland Raider offense, so we'll prepare for their offense, probably try to have an understanding of what he does well. They are both mobile; Derek Carr can really move, too. So, it's not like you're talking about two guys that aren't mobile. They're both mobile quarterbacks."
Del Rio initially said after the game that Carr was suffering from back spasms and did not believe the injury would be a long-term issue. But further tests on Monday revealed the nature of the injury.
"The MRI wasn't conclusive, so they did a CAT scan and that's where it was determined that there was a problem," Del Rio said. "He did the CAT scan after this morning, like this afternoon. This all just happened. The timeline was, first the X-ray yesterday, doesn't look like anything. Then the MRI this morning, still looks good. Then, let's be sure because we didn't have a real good look at this area, so the CAT scan revealed the problem."
Information from ESPN's Jamison Hensley and ESPN.com injury analyst Stephania Bell was used in this report.