The singer is hoping to get the 'Bad Romance' star to be the opening act for her next year, according to the News of the World.
A source said: "Madonna is absolutely obsessed with GaGa.
"She thinks she is the most cutting-edge star to come out of pop in ages and desperately wants to work with her. Basically, GaGa can name her price."
The report claims that plans for Madonna's next tour are already under way, as part of her deal with Live Nation.
