The couple were spotted enjoying a drink in Sheeran's hometown and have attended each other's concerts regularly, reported OK! magazine.
The "Thinking Out Loud" hitmaker, 24, considered himself to be in the "early stages of dating" with the Pussycat Doll, according to a close friend.
The flame-haired singer invited Scherzinger, 37, to be a guest at his three London Wembley Arena shows.
Scherzinger first sparked rumours of their closeness when she shared a photo with Sheeran backstage at his North London concert.