The 24-year-old had already signed a pre-contract deal with the Gers but made the move to Glasgow on deadline day after nearly six years as a professional with the Cherries.
The 6ft 4in centre-back has played just 35 times for Bournemouth, with a third of those appearances coming this season in the Championship.
He was capped once by England at Under-21 level and played for non-league side AFC Totton on loan in 2015.
Simpson told Rangers’ website: "I am super excited to be here. This is a massive club and I am honoured to be a part of it. I can't wait to pull on the shirt, and play in front of the fans and at the stadium.
"It is obvious how big this club is - it's one of the most successful clubs in world football so I am delighted to be here.
"It is going to be a huge challenge to keep up with everything that has happened and to keep pushing myself to stay at the standards that are expected here.
"That is very important to me and I am excited to try and do that."
Manager Steven Gerrard added: “We are delighted to have Jack here. We have been aware of him for some time and excited to finally have him.
“Clearly it’s a bonus to have him here now earlier than initially planned. His presence makes us stronger and we are all looking forward to working with him.”