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 | |
| 1 | Wycombe WanderersWycombe | 13 | 9 | 2 | 2 | 31 | 16 | 15 | 29 |
| 2 | Birmingham CityBirmingham | 12 | 9 | 2 | 1 | 23 | 11 | 12 | 29 |
| 3 | Wrexham | 13 | 7 | 4 | 2 | 21 | 10 | 11 | 25 |
| 4 | Mansfield TownMansfield | 12 | 7 | 3 | 2 | 20 | 14 | 6 | 24 |
| 5 | Lincoln CityLincoln | 13 | 7 | 3 | 3 | 20 | 16 | 4 | 24 |
| 6 | Bolton WanderersBolton | 13 | 7 | 2 | 4 | 22 | 19 | 3 | 23 |
| 7 | Huddersfield TownHuddersfield | 13 | 7 | 1 | 5 | 19 | 12 | 7 | 22 |
| 8 | Barnsley | 13 | 6 | 4 | 3 | 21 | 18 | 3 | 22 |
| 9 | Reading | 13 | 7 | 1 | 5 | 21 | 20 | 1 | 22 |
| 10 | Stockport CountyStockport | 14 | 5 | 6 | 3 | 20 | 17 | 3 | 21 |
| 11 | Exeter CityExeter | 12 | 6 | 1 | 5 | 13 | 10 | 3 | 19 |
| 12 | Charlton AthleticCharlton | 13 | 5 | 4 | 4 | 15 | 14 | 1 | 19 |
| 13 | Peterborough UnitedPeterborough | 14 | 5 | 3 | 6 | 25 | 26 | -1 | 18 |
| 14 | Rotherham UnitedRotherham | 14 | 4 | 5 | 5 | 14 | 16 | -2 | 17 |
| 15 | Stevenage | 14 | 5 | 2 | 7 | 10 | 15 | -5 | 17 |
| 16 | Blackpool | 13 | 4 | 4 | 5 | 21 | 25 | -4 | 16 |
| 17 | Bristol Rovers | 13 | 5 | 1 | 7 | 15 | 20 | -5 | 16 |
| 18 | Northampton TownNorthampton | 13 | 4 | 3 | 6 | 17 | 20 | -3 | 15 |
| 19 | Wigan AthleticWigan | 13 | 3 | 5 | 5 | 12 | 11 | 1 | 14 |
| 20 | Leyton Orient | 13 | 3 | 2 | 8 | 11 | 18 | -7 | 11 |
| 21 | Crawley TownCrawley | 14 | 3 | 2 | 9 | 12 | 25 | -13 | 11 |
| 22 | Cambridge UnitedCambridge | 12 | 3 | 1 | 8 | 11 | 17 | -6 | 10 |
| 23 | Shrewsbury TownShrewsbury | 14 | 2 | 2 | 10 | 13 | 25 | -12 | 8 |
| 24 | Burton Albion | 13 | 0 | 5 | 8 | 13 | 25 | -12 | 5 |
| > League One Full Table | |||||||||
