Martin, who split from Hollywood actress Gwyneth Paltrow in March after 10 years of marriage, started dating Lawrence, 24, in June, reported Us magazine.
"He writes her songs and loves taking her to private romantic spots. Chris is sweeping Jen off her feet. That's his thing," a source said.
Lawrence initially told friends that her romance with the 37-year-old rocker, who has daughter Apple, 10, and son Moses, eight, with Gwyneth, wasn't serious, but is becoming increasingly attracted to him since watching him perform on stage in the UK earlier this summer.