Hyndman, who made his U.S. debut at the age of 18, joined Bournemouth from Fulham as a free agent over the summer. He was injured in
preseason and never dressed for an EPL game.
His two other appearances for the Cherries before Sunday's shock 3-0 loss at third-tier Millwall in the FA Cup came in the League Cup.
Hyndman should see playing time as Rangers, second in its first season back in the Scottish Premier League, looks to reinforce its midfield. It is also expected to sign Spaniard Jon Toral on loan from Arsenal.
Hyndman, the grandson of former FC Dallas coach Schellas Hyndman, joined Fulham's academy in 2011 at the age of 15. German Felix Magath gave Hyndman his pro debut in 2014.
The Scottish Sun dug up some old quotes from Magath, touting Hyndman as "the new Andrea Pirlo."
“I have never seen a player of his age play like he does -- he passes like an old professional," Magath said in 2014. "I’m sure he will be a good player. His role could be similar to the one that Andrea Pirlo plays, responsible for the forward play of the team from deep.”
Hyndman has hoped regular playing time will get him back in the national team picture under new coach Bruce Arena.