The 19-year-old striker will stay with the League Two club until February 9.
"A lot of hard work has taken place to get this young lad to the club," manager Steve Evans told Rotherham's official website.
Morgan has made three appearances in the Europa League for Liverpool this season.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| 1 | Liverpool | 12 | 10 | 1 | 1 | 24 | 8 | 16 | 31 |
| 2 | Manchester CityMan City | 12 | 7 | 2 | 3 | 22 | 17 | 5 | 23 |
| 3 | Chelsea | 12 | 6 | 4 | 2 | 23 | 14 | 9 | 22 |
| 4 | Arsenal | 12 | 6 | 4 | 2 | 21 | 12 | 9 | 22 |
| 5 | Brighton & Hove AlbionBrighton | 12 | 6 | 4 | 2 | 21 | 16 | 5 | 22 |
| 6 | Tottenham HotspurSpurs | 12 | 6 | 1 | 5 | 27 | 13 | 14 | 19 |
| 7 | Nottingham ForestNott'm Forest | 12 | 5 | 4 | 3 | 15 | 13 | 2 | 19 |
| 8 | Aston Villa | 12 | 5 | 4 | 3 | 19 | 19 | 0 | 19 |
| 9 | Fulham | 12 | 5 | 3 | 4 | 17 | 17 | 0 | 18 |
| 10 | Newcastle UnitedNewcastle | 12 | 5 | 3 | 4 | 13 | 13 | 0 | 18 |
| 11 | Brentford | 12 | 5 | 2 | 5 | 22 | 22 | 0 | 17 |
| 12 | Manchester UnitedMan Utd | 12 | 4 | 4 | 4 | 13 | 13 | 0 | 16 |
| 13 | Bournemouth | 12 | 4 | 3 | 5 | 16 | 17 | -1 | 15 |
| 14 | West Ham UnitedWest Ham | 12 | 4 | 3 | 5 | 15 | 19 | -4 | 15 |
| 15 | Everton | 12 | 2 | 5 | 5 | 10 | 17 | -7 | 11 |
| 16 | Leicester CityLeicester | 12 | 2 | 4 | 6 | 15 | 23 | -8 | 10 |
| 17 | Wolverhampton WanderersWolves | 12 | 2 | 3 | 7 | 20 | 28 | -8 | 9 |
| 18 | Ipswich TownIpswich | 12 | 1 | 6 | 5 | 13 | 23 | -10 | 9 |
| 19 | Crystal Palace | 12 | 1 | 5 | 6 | 10 | 17 | -7 | 8 |
| 20 | Southampton | 12 | 1 | 1 | 10 | 9 | 24 | -15 | 4 |
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