The 19-year-old midfielder becomes Millers boss Redfearn’s second signing since arriving at the AESSEAL New York Stadium.
Barker, who has represented England at Under-18 level, only signed a five-year professional contract with City in August, after rising through the club’s Academy. He has signed a youth loan deal.
“He is an exciting young player,” said Redfearn.
“He represents a ball carrier who will go past players and make things happen.
“He is a goalscorer who is quite young in years but has had quite a lot of experience around the first team at Manchester City.
“They think that he is going to develop into a top player so for us to get him is a massive plus. It is a great deal for us, he is what we need.
“He has the ability to open sides up and he can create goalscoring opportunities and he can score goals. He will be a massive addition for us.”
United defender Frazer Richardson is hoping he has done enough to face his former club Ipswich Town today.
For the 33-year-old full-back was handed a rare start for the Millers in last week’s narrow 1-0 loss at Middlesbrough. It was his first start since August after finding opportunities limited under former boss Steve Evans.
Richardson – who had a short loan spell at the Tractor boys in 2014 – hopes he did enough against Boro to keep his place.
“I’m a hard-working lad so I’ve just gone into training every day, worked as hard as I’ve ever done and seen where that’s got me,” he said. “Luckily enough, it’s got me a start against Middlesbrough and hopefully it will now get me a few more starts.”