The 24-year-old Englishman was being pursued by several clubs in the English Championship but the Ibrox side appear to have won the race with the transfer expected to go through in the next 48 hours, subject to a medical.
Cantwell, who has four England U21 caps, was given permission to find a new club during the current transfer window with his contract at Carrow Road due to expire in the summer. He has played more than 120 games for Norwich and has two promotions to the Premier League under his belt.
Respected transfer source Fabrizio Romano tweeted: “Rangers and Norwich are now signing contracts for Todd Cantwell deal, it’s done – agreement in place on all the details.
“Cantwell will undergo medical tests and sign the contract within 48 hours if all goes to plan.”