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