Shaw was Division 1 Feminine's top scorer during the 2020-21 season, scoring 22 goals.
"What really attracted me was the way Manchester City plays. The way they build up, the way they use the channels, the way they play through the team, the channels and how exciting they are when they're going forward," Shaw told ESPN.
"For me, looking in from the outside watching that I thought to myself I could see myself playing there and I could see myself being very successful there."
Jamaica's Shaw joined Bordeaux in 2019 after playing her college football in the United States. She opted not to enter in the National Women's Soccer League draft and instead chose to pursue options outside of the U.S.
"The players that are playing there [the WSL] the different cultures, it is exciting every time you watch a game in England. It's why I decided to play there," she added.
"It took a lot of hard work and a lot of sacrifices to leave my home town and travel to Europe."
The 24-year-old picked out new teammate Lucy Bronze as someone she is excited to play alongside, having played against Bronze in France when she was with Lyon.
"To compete against her and now be on the same team as her, I think it will be a great experience because you could tell the sort of player she was. She's so motivated and she always wants to win," Shaw said.
"Playing in the same team, I think she could motivate me because she has a lot of experience under her belt and I'm open to learning and growing."
Shaw was part of the Jamaica squad that were the first Caribbean nation to qualify for a World Cup. The team are playing in the Summer Series in the U.S., but Shaw hasn't joined the squad with the French season finishing at the start of June. Jamaica beat Nigeria 1-0 and lost to the United States Women's National Team (USWNT) 4-0.
"Our team is a very small group. It's extremely difficult to compete at certain levels with the lack of investment we currently face," Shaw said.
"For me and my teammates, we just want to play and sometimes you need to look past certain situations. That's why I've got where I am.
"It was never easy for me. I lost a lot of relatives growing up. The way I played when I was growing up against boys was hard.
"We don't have a lot of camps and we don't play a lot of international friendlies but we qualified for the World Cup so I think having that motivation and that will to want to compete and be the best version of yourself helps you get to a certain part of life."