The Blades have been interested in the 20-year-old for a number of weeks but were forced to wait for him to return to fitness after a hamstring injury.
Riley played 85 minutes of United's 1-0 Premier League 2 victory over Liverpool on Monday.
The full-back has made two first-team appearances for Manchester United.
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Earlier this year we asked where the centre spots of various cities around England were - these being the points distances on road signs are measured to.
Google identifies the centre spot of Newcastle as being in a cafeteria in the King George VI building at Newcastle University, while the city council believed it might be Grey's Monument.
Philip James, a senior lecturer in geographical information science (GIS) at the university, has been doing some investigating, and the answer is fairly simple - it depends who you ask.
"Different providers give different answers depending upon what method they are using," said Mr James, who specialises in mapping people's use of places through their online activity, such as where they tweet from.
Facebook marks the centre point as being near Exhibition Park in the Town Moor, Michelin next to the Haymarket Metro station and the Ordnance Survey places it close to the Manors Metro station.
Twitter marks it as being on Nuns Moor in the north-west of the city while Open StreetMap places the centre point near Kenton in the north of Newcastle.
Why such a division of opinion?
"The Newcastle administrative area is not a simple shape so determining the centre is not easy," said Mr James.
"One method is to draw a boundary box around it then use the centre of that rectangle as your centre point.
"Of course that will give you a different centre to the methods others might use."
One answer that is probably as good as any other is to take the mean of all these various sources and see where that leads, which is what Mr James did.
This approach shows the centre point as being outside the front of the Civic Centre.
But if you ask people where they think it should be, which we did, they invariably gave the same answer as the council - Grey's Monument at the top of Grainger Street.
Comments varied from "it's the biggest thing and can be seen from anywhere" to "it's where the shops are".
And even Mr James said he would consider that the centre in any meaningful way.
"It's the point most people know and where they arrange to meet," he said.
Ultimately, does knowing the centre point even matter?
"I don't think it causes us any real issues in a practical sense," Mr James said.
"Most people will know where they are going, either from knowledge of the area or by searching for actual addresses, I can't really see any scenarios where knowing the absolute centre of Newcastle is very important. It's just an interesting question though.
"A lot of arbitrary decisions will have been made in the past, probably for good reasons at the time, and they will simply still be used today to determine things like distances for road signs and city centre spots."
League One leaders Sheffield United have signed Manchester United full-back Joe Riley on loan for the rest of the season.
[NEXT_CONCEPT]
An academic at Newcastle University has been taking a closer look at a question we posed a short while ago - where exactly is the centre point of Newcastle?