The USA international has impressed during a trial period with the ‘Gers.
Gerrard revealed last week that the club were hatching a long-term contract for the 25-year-old.
Polster will add versatility to the Rangers ranks seeing as he is comfortable at both right-back or in midfield.
The Chicago Fire ace is set to become their fifth signing of a busy January window.
Polster was part of the squad for the mid-season winter training camp in Tenerife and clearly impressed.
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Gerrard said last week: “We’re in talks with his agent. We haven’t got it over the line just yet, but we’ve told his advisers we’d like him to come.
“It would be a long-term deal. I know him well from the MLS. He’s a player who can play numerous positions.
“He’s a good professional, he’s in good shape and I think he’s got the potential to be a first-team player for us if the financials work.”
The former Liverpool man has dismissed talk that Rangers are in the hunt for Nick Powell.
There has been speculation that the club were set to go head-to-head with Old Firm rivals Celtic for his signature.
The former Manchester United man is out of contract at the end of the season and will be available on a free transfer.
When questioned whether there was any genuine interest, Gerrard said: “It’s the first I’ve heard of it.”
He did however confirm that goalkeeper Stefan Marinovic is currently on trial at Ibrox.
Rangers were due to take on Cowdenbeath in the 4th Round of the Scottish Cup this weekend however the match was postponed due to a frozen pitch.
They instead beat HJK Helsinki 3-2 in a friendly.
New signings Jermain Defoe and Steven Davis both started the match before being substituted at half time.