Davidson rang in the new year in Miami, Florida, where he co-hosted Miley Cyrus' New Year's Eve Party alongside the chart-topping pop star.
Perhaps the new couple feel the need to get some space from rumours that Kanye West - Kim's estranged husband - still "wants to win her back".
Kim - who has North, 8, Saint, 6, Chicago, 3, and Psalm, 2, with the rapper - filed for divorce last year, but West is desperate to rekindle their romance and he recently bought a $4.5 million home on the same street as the reality star.
A source said: "Kanye really wanted to buy a home near Kim and their kids to be close to them.
"Kanye saw a house on Kim's block was on the market and jumped at the chance to get it."
Kanye, 44, feels it will be "easier to co-parent with Kim if they live close to each other".
The brunette beauty is said to be "fine" with Kanye's decision to buy a property nearby, but she hasn't shown any interest in reigniting her romance with the musician.
Not one to be easily dissuaded, however, West is not giving up the fight ant time soon, with a source confirming: "Kanye believes he still has a chance".