Gerrard revealed Rangers have offered a long-term contract to 25-year-old Polster who is a free agent after turning down a new deal at Chicago Fire.
Polster would be the Ibrox club’s third recruit of the current window after the loan captures of striker Jermain Defoe and midfielder Steven Davis.
Rangers have been linked with a host of other players, including Gerrard’s former Liverpool team-mate Adam Lallana, Lazio midfielder Ravel Morrison and Derby County’s Scottish international midfielder Craig Bryson.
But Gerrard was dismissive of the speculation and insists Polster could be the only further addition to his squad before the end of the month.
“I don’t want to say a definite ‘no’ as something might happen in the coming weeks that we do or act upon,” he said. “But, at the moment, our business will be where it is now. I won’t say that’s 100 per cent the case and make a liar out of myself as things can change. There may be one more signing but don’t be surprised if there’s not.
“Loads of names have come up. I read about Lallana. Five or six have been bandied about. I know there’s been a bit of noise in the media [about Bryson] but it must be coming from Craig’s side as it’s not from our side.
“We are in talks with Matt Polster’s agent. We haven’t got it over the line just yet but we’ve told his advisers that 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 in numerous positions, a good professional in good shape and he has the potential to be a first-team player at Rangers.”
Gerrard confirmed that midfielder Jordan Rossiter, who has struggled to secure first-team football this season, will be allowed to move out on loan this month. Dundee United and Fleetwood Town have both been linked with Rossiter but Gerrard says he has received no firm offers for the player yet.
Defender Joe Worrall is expected to remain at Rangers for the rest of the season with Gerrard stating his parent club Nottingham Forest have missed an agreed deadline of 7 January in which they could have recalled him.
Gerrard would not be drawn on whether Rangers will attempt to advance the signings of Kilmarnock winger Jordan Jones and Dundee midfielder Glen Kamara, who have both agreed pre-contract deals to move to Ibrox in June.
But he was adamant that midfielder Ross McCrorie, whom he described as a “future captain” of Rangers, will not be allowed to join Dundee on loan as had been speculated this week.
McCrorie will miss Rangers’ Scottish Cup fourth round tie at Cowdenbeath tonight through suspension, while defender Connor Goldson is also set to miss out because of injury.
Gerrard, meanwhile, scotched rumours he is poised to make a playing return in Rangers’ friendly match against HJK Helsinki at Ibrox on Sunday, a fixture agreed as part of striker Alfredo Morelos’ transfer from the Finnish champions.
“Where has that come from?” said 38-year-old Gerrard who hung up his boots two years ago. “I’d love to play again but unfortunately it won’t be happening.”