The `Fool`s Gold` actress - who was most recently romantically linked with baseball star Alex `A-Rod` Rodriguez - has reportedly enjoyed a string of dates with the Muse frontman but has been trying desperately to keep their romance under wraps.
A source close to the British rocker said: "Matt has been over to America to hang out with Kate.
"But they have done everything they can to avoid being seen together in the early stages of their relationship."
The couple were spotted having dinner together in a New York eatery earlier this week, but plan to make their first official pubic appearance together at the Glastonbury festival - which Muse are headlining - in Somerset, South England, later this month.
The source added to The Sun newspaper: "Things are going so well Kate is planning to fly to the UK to hang out with Matt at the festival.
"She wants to watch the band from the side of the stage and meet his pals."
Kate, 31, was previously married to the Black Crowes rocker Chris Robinson, with who she has a six-year-old son Ryder.
Matt, 32, split from his fiancée, Italian psychologist Gaia Poloni late last year.