Now Southampton boss Harry Redknapp has revealed that his club are also in the race to sign the 26-year-old Republic of Ireland international forward.
"I am hoping we still have a chance of getting him," Redknapp told his club's official website after the Saints 3-0 Carling Cup, 1st Round win against Southend on Monday evening.
"The chairman is working on it and that's all you can ask. I believe we have made a similar offer to Palace."
"I would not say we were favourites.
"Norwich are the only ones who have agreed a fee but I don't think he wants to go there, or else they could not settle terms.
"I have not spoken to him yet because we have not had a bid accepted so I am not allowed to talk to him. But I am hoping that will change."
Morrison joined Blues from Palace three seasons ago for £4.25 million with striker Andrew Johnson moving to Selhurst Park as part of the deal.
However, he has failed to hold down a regular starting place at St Andrew's and wants to leave to secure regular first team football.