Steven Gerrard's side confirmed the move for the 25-year-old, which is subject to international clearance, on Wednesday.
Polster, who can also play at right back, sufficiently impressed Gerrard on trial at Ibrox to become Rangers' fourth signing of the January transfer window, joining Steven Davis, Jermain Defoe and Andy Firth.
"This is something that I’ve always dreamed about doing," Polster said.
"As a little boy, a lot of people would know that I wanted to play in Europe, before even going to MLS.
"To finally do it, for this kind of club has left me feeling speechless in a way.
"Obviously here, the standards and expectations are high and that's what I want to be part of."
Polster won his only cap to date for the United States in January 2018, and has played in Chicago since he was drafted seventh overall by the Fire in the 2015 MLS SuperDraft, but only made three league appearances last season.
He will wear the number 25 jersey and joins a side sitting three points behind defending champions Celtic at the top of the Scottish Premiership.