The `Fool`s Gold` actress and the British rock star are looking for a property in New York after dating for just four months.
The couple have been inseparable since meeting at the Coachella Festival in California earlier this year and believe the time is right to set up home.
A source told The Sun newspaper: "They`re very loved-up, they just click.
"Kate`s had a place in the city for five years but she wants somewhere bigger, so Matt`s suggested pooling their resources."
Kate wants the home to have a playroom for her six-year-old son Ryder - her child from her marriage to Black Crowes singer Chris Robinson -while Matt needs a studio to make music.
The source added: "Kate wants a studio for Matt and playroom for Ryder."
Despite being snapped in public at the start of this year at a New York eatery, Kate and Matt decided to make their first official public appearance at the Glastonbury music festival in June - at which Muse was headlining.
The 31-year-old actress watched her man perform from the side of the stage and the loved-up pair spent the rest of their time at the famous event partying with pals.
Matt, 32, also met Kate`s mother, actress Goldie Hawn, earlier this month and admitted he was "quite nervous" beforehand.
And he also revealed marriage wasn`t out of the question.
He said: "We just have fun together and are seeing how it goes. Kate is great.
"It`s a little bit too soon to talk about weddings and stuff. Ask me in a few months though. I`ll let you know how it goes with Goldie - and then who knows?"