US R&B singer Jesse Braham has claimed in legal papers Swift stole the words from a song he wrote in 2013 called Haters Gone Hate.
As well as the monetary compensation, Mr Braham also wants his name added as a writer on the track.
Representatives for Swift have yet to officially comment on the legal case.
Shake It Off topped music charts around the world and reached number two in the UK. The video for the song has been watched more than 1.1 billion times on YouTube.