Khloe and her NBA superstar BF spent their Wednesday house hunting in L.A. We're told they're not buying a house together ... he's looking to rent a place in the Hollywood Hills or the Valley where he and Khloe can stay when they're in the city. She has a house in Calabasas -- Bieber's old house -- about an hour's drive from Hollywood..

We're told Tristan's budget is around $60k a month, and he wants to rent through September.

Khloe's said she'd marry Tristan if he proposed ... maybe this is the first step.