The pair were spotted getting friendly at an intimate gig in Nashville last month and have reportedly been spending time together since in the 24-year-old singer's London home, as well as enjoying a break together in the Orkney Islands in Scotland last week.
A source told The Daily Mail newspaper: "Marcus invited Carey to watch him play a secret show in the basement of his friend Mike Harris's home in Nashville. She sat in the front row and while he was singing Marcus looked down at Carey and she blushed. It was clear they had chemistry.
"Afterwards they went to the City House restaurant and dined with friends but they sat together and acted like a couple."
Now the 25-year-old actress - who has previously dated 'Transformers' star Shia LeBoeuf and was last month linked to actor Eddie Redmayne - is said to want to see even more of her new beau and may even accompany him and his band when they tour the US.
The source said: "Carey's hoping they can spend more time together when Marcus flies back to America next month for a major tour with his band.
"He's hired a vintage train instead of a tour bus and has asked Carey to join him for the trip. It's early days but they seem to enjoy each other's company."