Tristan is said to be looking for a property in the upmarket areas of Bel Air, Beverly Hills or the Hollywood Hills so they can have a base near the city as well as Khloe's Calabasas pad.
Mail Online has contacted Tristan's representatives for comment.
She then accessorised her look with lace up boots and a leather Hermes Birkin handbag.
And on Saturday, Khloe Kardashian appeared to conceal her stomach as she stepped out with boyfriend Tristan Thompson in Cleveland, Ohio, rocking matching his 'n' hers hoodies.
The host of Revenge Body added a pair of high-heeled black lace-up boots and sunglasses as she spent time with her Cleveland Cavaliers beau.
Khloe and younger boyfriend Tristan, 26, were first photographed together in early September 2016 in Mexico.
The former LA Lakers star wed Khloe in 2009 but they separated four years later with Khloe filing for divorce in December 2013.
However, following Lamar's overdose and collapse at a Nevada brothel in July 2015, she put the divorce on hold temporarily in order to be able to take charge of his medical care.
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