The 25-year-old Irishman, who joined the Dons from Wolverhampton Wanderers in 2009, is out of contract this summer.
While a number of clubs are thought to be monitoring Gleeson, Birmingham are confident of winning the race for his signature, according to Sky Sports News.
Gleeson scored three goals in 42 appearances for MK Dons during the 2013-14 campaign.
Restore Data
| Team | P | W | D | L | F | A | GD | PTS | |
| C | Leicester CityLeicester | 45 | 31 | 4 | 10 | 89 | 39 | 50 | 97 |
| 2 | Ipswich TownIpswich | 45 | 27 | 12 | 6 | 90 | 57 | 33 | 93 |
| 3 | Leeds UnitedLeeds | 45 | 27 | 9 | 9 | 80 | 41 | 39 | 90 |
| 4 | Southampton | 45 | 25 | 9 | 11 | 85 | 62 | 23 | 84 |
| 5 | Norwich CityNorwich | 45 | 21 | 10 | 14 | 79 | 63 | 16 | 73 |
| 6 | West Bromwich AlbionWest Brom | 45 | 20 | 12 | 13 | 67 | 47 | 20 | 72 |
| 7 | Hull City | 45 | 19 | 13 | 13 | 68 | 59 | 9 | 70 |
| 8 | Middlesbrough | 45 | 19 | 9 | 17 | 68 | 61 | 7 | 66 |
| 9 | Coventry CityCoventry | 45 | 17 | 13 | 15 | 69 | 57 | 12 | 64 |
| 10 | Preston North EndPreston | 45 | 18 | 9 | 18 | 56 | 64 | -8 | 63 |
| 11 | Bristol City | 45 | 17 | 11 | 17 | 53 | 47 | 6 | 62 |
| 12 | Cardiff CityCardiff | 45 | 19 | 5 | 21 | 51 | 65 | -14 | 62 |
| 13 | Swansea CitySwansea | 45 | 15 | 12 | 18 | 59 | 64 | -5 | 57 |
| 14 | Watford | 45 | 13 | 17 | 15 | 60 | 58 | 2 | 56 |
| 15 | Sunderland | 45 | 16 | 8 | 21 | 52 | 52 | 0 | 56 |
| 16 | Millwall | 45 | 15 | 11 | 19 | 44 | 55 | -11 | 56 |
| 17 | Queens Park RangersQPR | 45 | 14 | 11 | 20 | 45 | 57 | -12 | 53 |
| 18 | Stoke CityStoke | 45 | 14 | 11 | 20 | 45 | 60 | -15 | 53 |
| 19 | Blackburn RoversBlackburn | 45 | 13 | 11 | 21 | 58 | 74 | -16 | 50 |
| 20 | Sheffield WednesdaySheff Weds | 45 | 14 | 8 | 23 | 42 | 68 | -26 | 50 |
| 21 | Plymouth ArgylePlymouth | 45 | 12 | 12 | 21 | 58 | 70 | -12 | 48 |
| 22 | Birmingham CityBirmingham | 45 | 12 | 11 | 22 | 49 | 65 | -16 | 47 |
| 23 | Huddersfield TownHuddersfield | 45 | 9 | 18 | 18 | 48 | 75 | -27 | 45 |
| R | Rotherham UnitedRotherham | 45 | 4 | 12 | 29 | 32 | 87 | -55 | 24 |
| > Championship Full Table | |||||||||
