The couple have reportedly put a bid on a nine-bedroom home in the Hollywood Hills, according to TMZ.
Minaj has been without a permanent base in the city since her lease ran out on the property she shared with ex boyfriend Safaree Samuals, who she split from in August 2014 after 12 years together, and Mill is said to be keen to move from his native Philadelphia to Los Angeles.
Minaj, 32, and Mill, 28 were first linked together earlier this year but things are said to have already gotten extremely serious between them.