The Formula 1 driver - who got back together with the 'Right There' singer over Christmas after they split in July 2013 - is trying to persuade her to swap London life for Monaco and move into his luxury home.
A source told The Sun newspaper: "Nicole is giving serious thought to moving in with Lewis.
"Both sets of families are behind them being back together and she and Lewis say it feels right.
"They know they can't keep splitting up and getting back together, so more commitment is the next step."
The on/off couple, who started dating in 2007 after meeting at the MTV Europe Music Awards in Berlin, are said to be back on track and Lewis, 29, has promised the 'X Factor' judge, 35, that she can use his private jet to go to London whenever she wants.
The couple spent New Year together in Aspen, Colorado and stepped out in public for the first time in seven months in Los Angeles, California last Saturday (11.01.14).
Lewis bought a home in Monaco after moving from Geneva in 2012 so he could be close to his Formula One events.
The source added: "After reuniting at Christmas, Lewis has decided he wants Nicole at his side for good.
"Nicole is always travelling all over the world with work, but rather than LA and London, Lewis wants their base to be in Monaco.
"They can go about there and no one pays them any attention, that's what he likes about the place.
"Nicole is mulling things over. They have been talking a lot about the future."