The 23-year-old was a big part of Nigel Adkins' plans for the upcoming season, but a bid was made on Wednesday afternoon and the centre-back travelled to south Wales on Thursday to agree personal terms.
The Chronicle understand that the Bluebird paid an initial £3 million for the defender, who made 45 appearances for the Royals since joining from Swindon Town in 2011.
The fee also includes a number of add-ons based on promotion and appearances, which could net the Royals further cash in the future.
Morrison signed a new four-year deal at the Madejski Stadium in 2013 but it seems the lure of signing for the side that many consider to be favourites for the title was too good to turn down for the Plymouth-born defender.
With Anton Ferdinand still a couple of weeks away from match fitness, it is thought that Michael Hector will partner Alex Pearce at centre-back for the visit of Ipswich Town on Saturday.
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