The Ibrox club have sealed a pre-contract agreement for the defender, who has long been admired by boss Steven Gerrard.
The 24-year-old first attracted the manager’s interest two years ago but the link re-surfaced this month and Cherries boss Jason Tindall revealed Rangers had discussed a move with the player.
After making his mind up, Simpson has agreed to move north, despite interest from Cardiff City and Sheffield Wednesday. Rangers have also been linked with south coast team-mate Nnamdi Ofoborh.
The club confirmed the deal this morning, and sporting director, Ross Wilson, said: “Jack is a player that Steven and I have discussed a number of times in recent months.
“We are always looking at how we can improve and Jack fits the profile that we are looking to add as we develop and move the squad forward again.
“His contractual situation at Bournemouth provided a natural and attractive moment for us to pursue Jack’s signature. It has been well publicised that he has been a much-coveted player south of the border but from our first discussion with Jack, we could see how the opportunity to join Rangers was one that was very appealing to him.
“The situation with pre-contract negotiations is never an easy one and I extend my thanks to Richard Hughes, my counterpart at Bournemouth, for how he has handled our interest in Jack.”