Tabloid reports claim that the 24-year-old met up with Cooper at Scotts in Mayfair earlier this week for a date. Allen's friend Damien Hirst also attended the meal, according to The Sun.
"Lily is tired of only ever being involved with well-known blokes," said a source. "She wanted to go out with someone her own age who isn't involved in music or the media.
"Sam is really down-to-earth. He grafts hard for a living as a painter and decorator and leads a totally normal life. Lily is completely smitten with him."
Allen's previous lovers include art dealer Jay Jopling, Chemical Brothers DJ Ed Simons and record company worker Seb Chew.