The Ghanaian has been a key figure in the Robins' lineup following his £150,000 move from Barnet two years ago.
According to The Mirror, Swans boss Michael Laudrup is keen on adding the 24-year-old to his squad ahead of the new Premier League season.
Adomah, who made his international debut against Brazil at Craven Cottage last year, has scored 10 goals in 91 appearances for Bristol City.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| 1 | Leicester CityLeicester | 44 | 30 | 4 | 10 | 86 | 39 | 47 | 94 |
| 2 | Leeds UnitedLeeds | 44 | 27 | 9 | 8 | 80 | 37 | 43 | 90 |
| 3 | Ipswich TownIpswich | 43 | 26 | 11 | 6 | 85 | 53 | 32 | 89 |
| 4 | Southampton | 44 | 25 | 9 | 10 | 85 | 61 | 24 | 84 |
| 5 | West Bromwich AlbionWest Brom | 44 | 20 | 12 | 12 | 67 | 44 | 23 | 72 |
| 6 | Norwich CityNorwich | 44 | 21 | 9 | 14 | 77 | 61 | 16 | 72 |
| 7 | Hull City | 44 | 19 | 12 | 13 | 65 | 56 | 9 | 69 |
| 8 | Coventry CityCoventry | 43 | 17 | 12 | 14 | 68 | 55 | 13 | 63 |
| 9 | Middlesbrough | 44 | 18 | 9 | 17 | 64 | 60 | 4 | 63 |
| 10 | Preston North EndPreston | 44 | 18 | 9 | 17 | 56 | 61 | -5 | 63 |
| 11 | Cardiff CityCardiff | 44 | 19 | 5 | 20 | 50 | 61 | -11 | 62 |
| 12 | Bristol City | 44 | 16 | 11 | 17 | 51 | 47 | 4 | 59 |
| 13 | Sunderland | 44 | 16 | 8 | 20 | 52 | 51 | 1 | 56 |
| 14 | Swansea CitySwansea | 44 | 15 | 11 | 18 | 57 | 62 | -5 | 56 |
| 15 | Watford | 44 | 12 | 17 | 15 | 59 | 58 | 1 | 53 |
| 16 | Millwall | 44 | 14 | 11 | 19 | 43 | 55 | -12 | 53 |
| 17 | Stoke CityStoke | 44 | 13 | 11 | 20 | 44 | 60 | -16 | 50 |
| 18 | Queens Park RangersQPR | 44 | 13 | 11 | 20 | 41 | 57 | -16 | 50 |
| 19 | Blackburn RoversBlackburn | 44 | 13 | 10 | 21 | 58 | 74 | -16 | 49 |
| 20 | Plymouth ArgylePlymouth | 44 | 12 | 12 | 20 | 58 | 69 | -11 | 48 |
| 21 | Sheffield WednesdaySheff Weds | 44 | 13 | 8 | 23 | 39 | 68 | -29 | 47 |
| 22 | Birmingham CityBirmingham | 44 | 12 | 10 | 22 | 48 | 64 | -16 | 46 |
| 23 | Huddersfield TownHuddersfield | 44 | 9 | 17 | 18 | 47 | 74 | -27 | 44 |
| R | Rotherham UnitedRotherham | 44 | 4 | 12 | 28 | 32 | 85 | -53 | 24 |
| > Championship Full Table | |||||||||
