Norwich will pay £3million plus an extra £1m in add-ons to sign the Scot permanently in the summer.
Dorrans told the Championship club's official website: "I'm glad to finally get it sorted.
"I came here on the last day of the transfer window and for whatever reason it didn't work out. I've been in discussions ever since then, so although I was frustrated it never got done on the final day, hopefully now I can add to a good squad.
"I spoke to the manager (Alex Neil) and a couple of the boys, and they had nothing but nice things to say about the club.
"The manager and everyone here is working to get this club back to the Premier League, and I'm excited to be here.
"With the fans right behind us, hopefully we can do that as we've got some big games coming up. I'm looking forward to getting started, and to helping us push on."
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