Liverpool striker Luis Suarez has signed a new long-term contract with the club.
Malky Mackay looks set to take charge of Cardiff for the last time at Liverpool on Saturday.
The 18-year-old, who has played three times for the Reds in the Europa League this season, will join up with the Millers until February 9.
Boss Steve Evans revealed he has been after his man since the summer and is grateful to the Merseysiders for allowing Morgan to drop to the bottom tier.
"He is somebody who has been sought after many times buy many clubs, but Liverpool have kept him in house," Evans said.
"He has played in their first team and he has got a great goal record. He is someone I have admired and it's fair to say their manager will want to delete my number because I have been barraging him since early December.
"I've had a conversation with Brendan Rodgers and I've told him how grateful I am that he has allowed him to come and play at this level, because the level we play at was their concern.
"To be fair everything has played a part, our stadium, our attendance, our results and I have just kept asking. I have seen him seven or eight times and we have watched him closely since I tried to get him in the summer."