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 | 7 | 6 | 0 | 1 | 13 | 2 | 11 | 18 |
| 2 | Manchester CityMan City | 7 | 5 | 2 | 0 | 17 | 8 | 9 | 17 |
| 3 | Arsenal | 7 | 5 | 2 | 0 | 15 | 6 | 9 | 17 |
| 4 | Chelsea | 6 | 4 | 1 | 1 | 15 | 7 | 8 | 13 |
| 5 | Aston Villa | 6 | 4 | 1 | 1 | 12 | 9 | 3 | 13 |
| 6 | Newcastle UnitedNewcastle | 7 | 3 | 3 | 1 | 8 | 7 | 1 | 12 |
| 7 | Fulham | 7 | 3 | 2 | 2 | 10 | 8 | 2 | 11 |
| 8 | Tottenham HotspurSpurs | 6 | 3 | 1 | 2 | 12 | 5 | 7 | 10 |
| 9 | Brentford | 7 | 3 | 1 | 3 | 13 | 13 | 0 | 10 |
| 10 | Brighton & Hove AlbionBrighton | 6 | 2 | 3 | 1 | 10 | 8 | 2 | 9 |
| 11 | Nottingham ForestNott'm Forest | 6 | 2 | 3 | 1 | 6 | 5 | 1 | 9 |
| 12 | West Ham UnitedWest Ham | 7 | 2 | 2 | 3 | 10 | 11 | -1 | 8 |
| 13 | Bournemouth | 7 | 2 | 2 | 3 | 8 | 10 | -2 | 8 |
| 14 | Manchester UnitedMan Utd | 6 | 2 | 1 | 3 | 5 | 8 | -3 | 7 |
| 15 | Leicester CityLeicester | 7 | 1 | 3 | 3 | 9 | 12 | -3 | 6 |
| 16 | Everton | 7 | 1 | 2 | 4 | 7 | 15 | -8 | 5 |
| 17 | Ipswich TownIpswich | 7 | 0 | 4 | 3 | 6 | 14 | -8 | 4 |
| 18 | Crystal Palace | 7 | 0 | 3 | 4 | 5 | 10 | -5 | 3 |
| 19 | Southampton | 7 | 0 | 1 | 6 | 4 | 15 | -11 | 1 |
| 20 | Wolverhampton WanderersWolves | 7 | 0 | 1 | 6 | 9 | 21 | -12 | 1 |
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