The 25-year-old Canadian star arrived in Britain earlier this month on his Club Paradise trek, and hired Ora to be his support act.
The pair has already worked together on several tracks, but their professional relationship has blossomed into a romance, reported Sun online.
They were also seen kissing backstage at a gig in Newcastle.
"Rita and Drake have been good friends for a while now but things seemed more serious this week. They were all over each other backstage at the gig on Monday and weren't trying to hide it," a source said.
Drake has previously dated R&B star Rihanna, while Ora was linked to Rob Kardashian. PTI JCH