The couple recently reconciled after the brunette beauty admitted cheating on him with her Snow White and the Huntsman director Rupert Sanders last summer.
A source told the Sunday Mirror newspaper: “Ever since Kristen’s affair last year, Rob has realised just how much his friends and family in the UK mean to him.
“After spending more time here in recent months, Rob’s decided he wants to split his time more between London and Los Angeles.
“He wants to put down roots in London and he’s doing it with Kristen.
“He’s told her that if she wants to make things work, she’s going to have to accept that he wants to spend much more time in London.”
The couple already have their heart set on one luxury Victorian house in South West London, which has an "LA feel" to it and is close to Pattinson's parents' home.
The source added: "It’s on the market for millions. Unlike the others they saw, this is suited to their tastes and needs very little work.
“With their hectic filming schedules they want somewhere they can live in straight away and it has a beautiful garden too.
“They both fell in love with the property as soon as they saw it.
"They decided to base themselves in Richmond, South West London, so Rob can be close to his parents’ home in Barnes. Rob knows the area as he attended Tower House, a prep school near Richmond Park. So his dad Richard put him in touch with an estate agent who lined up appointments.”
Stewart, 22, is said to be "really excited" about spending more time in the UK and is even eyeing up a role in a West End production.
The source explained: “Kristen’s really excited about the prospect of spending more time in the UK and has been saying how she’d love to star in a West End show."