The disagreement began when Banks, 24, took objection to 23-year-old Malik's new "Like I Would" video, which was released on Tuesday.
She posted a picture to Instagram, pointing out the alleged similarities to her previous videos and style choices, reported NME magazine.
"Damn Zayn be mood boarding the ... Of out me," she wrote, adding, "I'm not mad about this though. Zayn is a cutie pie".
Malik later seemed to address Banks' criticism, writing: "No lies... I see you reaching but I don't care".
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"dude, I make better music than you. Simmer down with that fake white boy rebellion and that wannabe beiber swag," Banks further wrote.
"@zaynmalik imma start calling you punjab you dirty...