There are no competition concerns on the vast majority of routes under the franchise, the CMA said.
However, FirstGroup already operates the Great Western Railway (GWR) franchise, which runs the only other rail service between London and Exeter, and would therefore have control of both services, the CMA said.
CMA chief executive Andrea Coscelli, said: "This is a crucial rail route to the South West, used by around half a million passengers a year. It’s therefore vital that passengers do not suffer as a result of reduced competition."
"The CMA believes that without its intervention, FirstGroup may be able to increase fares for passengers between London and Exeter, as it will be the only rail operator running all services on this route. We look forward to hearing from the companies involved about how they think they can overcome this," Coscelli said.
The CMA has invited FirstGroup and MTR to propose steps to address the competition concerns. If those proposals are not acceptable, the CMA will begin an in-depth investigation, it said.
In previous cases, such as a decision on the award of the Northern franchise to Arriva, competition concerns have been resolved by the operator agreeing to price caps on the rail lines concerned, the CMA said.