Argentinian forward Becchio, who has scored 19 goals for Leeds this season, handed in a transfer request at Elland Road last week after talks over a new deal faltered.
Norwich hope the deal will be completed in time for Becchio, 29, to feature at QPR on Saturday.
A club statement read: "Becchio has agreed a three-and-a-half year deal with the Canaries' ending a five-and-a-half-year stint at the Yorkshire club, joining fellow former Leeds men Jonny Howson, Robert Snodgrass and Bradley Johnson in the City ranks."
Former Boca Juniors player Becchio feels he is ready to make the step up to the challenges of the English top flight.
He told Canaries Player: "I am really happy about having the chance to play in the Premier League, it is brilliant for me personally and for my career.
"The Premier League has some brilliant players including people like (Sergio) Aguero and (Carlos) Tevez (Becchio's compatriots) and it will be a great experience for me to play against players of that quality."
Wales striker Morison, meanwhile, will look to kickstart his career with Leeds after falling down the pecking order at Norwich after his £2.8m move in the summer of 2011.