The 21-year-old has scored twice in six appearances for the Giallorossi since his 2015 move from Serie B side Spezia.
Sadiq has spent the majority of his Roma career out on loan and has enjoyed spells at Bologna and Torino, as well as Dutch side NAC Breda.
"The plan was to stay in Rome this season but when Steven Gerrard calls, you can't say no," the Nigeria Under-23 international told Rangers TV.
"He told me about the team, the players, that everyone will be at my side if I come here and he will help me to go forward in my career.
"You don't have to think twice when someone says that to you."
Sadiq, who won a bronze medal at the 2016 Olympics, is the Light Blues' ninth summer signing.