Sources tell us the relationship is "just starting." The two were photographed on their way out of a movie date to see "Hunger Games" in NYC last night (above).
Apparently, Kanye has been after Kim for a while -- he just released a new song, "Theraflu" ... in which he says he had feelings for the reality star while she was dating Kris Humphries.
"And I admit I fell in love with Kim ... 'Round the same time she fell in love with him ... That's cool, babygirl, do your thing ... Lucky I ain't had Jay drop him from the team."
Kanye is referring to Jay Z -- who owns the NJ Nets ... the team Humphries plays for.
11:45 AM PDT -- Kim and Kanye just hit up a fancy restaurant on the UES for date #2 -- we got footage.
10:28 AM PDT -- So ... just a few hours ago, Kim was seen leaving from the building where Kanye lives ... wearing the same clothes she rocked to the movies last night.