The 30-year-old midfielder arrived in Glasgow earlier this week as Russell Martin bolstered his squad with a player fresh off back-to-back Premier League promotions at Southampton and Leeds United.
Speaking to the media for the first time as a Rangers player, Rothwell said of his excitement at the move: "I'm delighted. It probably took a little bit longer than I would have liked, but I'm delighted it's finally done and I can't wait to get going.
"You can see from the outside how huge the club is. It's built to win trophies.
"Hopefully, if the manager implements how he wants to do things, it will be successful and we can go on and win things.
"Just before I signed for Bournemouth, it was a toss-up between the two [clubs]. Selfishly, when I left Manchester United as a kid I always wanted to get back to the Premier League.
"They gave me the opportunity to do that and it's something I couldn't turn down at the time.
"I decided to go to Bournemouth, but I'm delighted that the offer came back around this time and I jumped straight at it.
Rothwell worked under Head Coach, Russell Martin, during his spell at Southampton in the second half of the 2023/24 campaign.
Together, they helped lift the south coast outfit back into the Premier League after a memorable play-off triumph over Leeds United at Wembley.
The midfielder has nothing but fond memories of working under Martin at the Saints and believes his understanding of the former Scotland international's beliefs about the game will be key on the training pitch.
He added: "There was a little bit of interest before the manager got the job, so as soon as I found out he was getting it he gave me a phone call and said I would be straight there.
"He sees the game how I would like to play the game if I was a manager. That was the main pull.
"But I've said before it's more about what he's like as a guy that you want to come and work for him.
"I can try and help the lads as I understand already how he wants to do things. He's only managed to work with the lads for a couple of weeks.
"If I can come in and help with stuff on the pitch like that then I'm sure it's something he'll lean on as well.
"He is a possession-based manager. From the outside it can look as if you're playing pointless passes, but when you're inside of it you understand every pass that you make is key.
"As long as we understand that as a team and the lads buy into it then I'm sure they'll fully enjoy the style of play.
"Before I came he ran me through how he sees me playing in the side. I already sort of knew what my role would be coming in here, but just to hear what he wants me to do and play it was an immediate yes for me.
"We have a really young squad at the moment, so I think that was why they wanted me to come in with a bit of experience.
"I'm really impressed by what I see so far and hopefully we can have a really successful season together."
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