The 19-year-old, who is widely regarded as one of the brightest talents in the Football League, has been linked with a number of top-flight clubs.
However, according to the South Wales Evening Post, the Swans have won the race for the versatile teenager.
It is believed that Swansea will part with £2m for the highly-rated midfielder.
Grimes has scored four times for Exeter this season.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| C | Portsmouth | 44 | 27 | 13 | 4 | 75 | 39 | 36 | 94 |
| 2 | Derby CountyDerby | 44 | 26 | 8 | 10 | 75 | 37 | 38 | 86 |
| 3 | Bolton WanderersBolton | 44 | 24 | 11 | 9 | 81 | 48 | 33 | 83 |
| 4 | Peterborough UnitedPeterborough | 43 | 24 | 8 | 11 | 84 | 56 | 28 | 80 |
| 5 | Barnsley | 44 | 21 | 12 | 11 | 79 | 60 | 19 | 75 |
| 6 | Oxford UnitedOxford Utd | 44 | 21 | 10 | 13 | 76 | 54 | 22 | 73 |
| 7 | Lincoln CityLincoln | 44 | 19 | 14 | 11 | 63 | 37 | 26 | 71 |
| 8 | Blackpool | 44 | 20 | 10 | 14 | 60 | 43 | 17 | 70 |
| 9 | Stevenage | 44 | 18 | 13 | 13 | 54 | 44 | 10 | 67 |
| 10 | Leyton Orient | 44 | 17 | 11 | 16 | 50 | 53 | -3 | 62 |
| 11 | Northampton TownNorthampton | 44 | 17 | 8 | 19 | 55 | 63 | -8 | 59 |
| 12 | Wycombe WanderersWycombe | 43 | 15 | 13 | 15 | 55 | 53 | 2 | 58 |
| 13 | Exeter CityExeter | 44 | 16 | 10 | 18 | 43 | 58 | -15 | 58 |
| 14 | Bristol Rovers | 44 | 16 | 9 | 19 | 52 | 64 | -12 | 57 |
| 15 | Wigan AthleticWigan | 44 | 18 | 10 | 16 | 59 | 55 | 4 | 56 |
| 16 | Charlton AthleticCharlton | 44 | 11 | 19 | 14 | 63 | 63 | 0 | 52 |
| 17 | Reading | 44 | 15 | 11 | 18 | 63 | 65 | -2 | 50 |
| 18 | Shrewsbury TownShrewsbury | 44 | 13 | 8 | 23 | 33 | 63 | -30 | 47 |
| 19 | Cambridge UnitedCambridge | 43 | 12 | 10 | 21 | 38 | 59 | -21 | 46 |
| 20 | Burton Albion | 44 | 11 | 10 | 23 | 36 | 62 | -26 | 43 |
| 21 | Cheltenham TownCheltenham | 43 | 11 | 8 | 24 | 37 | 61 | -24 | 41 |
| 22 | Port Vale | 44 | 10 | 10 | 24 | 41 | 72 | -31 | 40 |
| 23 | Fleetwood TownFleetwood | 44 | 8 | 13 | 23 | 45 | 72 | -27 | 37 |
| R | Carlisle UnitedCarlisle | 44 | 7 | 9 | 28 | 40 | 76 | -36 | 30 |
| > League One Full Table | |||||||||
