A deal worth up to £16m is expected to be completed by the end of the day.
The Gers rejected two offers in the summer for the 20-year-old who made his debut in January 2020.
Rangers will earn a club record initial fee of around £12m, with further add-ons worth around £4m, after they received a third move for Patterson early in the transfer window.
James Tavernier is the club's first-choice right-back, and he has started in just two of his six Scottish Premiership appearances this season.
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In October, he set up Rangers' goal in a win over Hibs.
He scored his first goal for the national side in November after making his Scotland debut in June.
The signing of Vitaliy Mykolenko from Dynamo Kiev was the reason for the move for Patterson.
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He scored his first goal for Scotland.
In October, when asked about the initial interest of the team, he said he was only interested in playing football.
"That's the thing that comes with it and I am not really aware of anything that happens, I just go on the pitch and do my thing and let the outside talk."
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In October, he said he was focused on his football.
I have been concentrating on my football and keeping my feet grounded.
You have everyone's opinions in football, and media everywhere.
Sometimes you need to surround yourself with the right people, and I feel like I have done that. I have a good group of friends, good team-mates and family who push me on to do well.
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