The English Premier League newboys had an offer of £12 million plus an additional £3 million in add-ons for the centre-back rejected last weekend.
John Percy, Midlands reporter for the Daily Telgraph, now reports the Bees have returned with an improved offer – thought to be £13.5 million with additional payments – which has been accepted by the Scottish Premiership club.
The Norwegian international has stated his desire to leave Celtic Park this summer following five successful years at the club he joined from Start in 2016.
Ajer is already in the final year of his contract at Celtic and would be able to walk away for free next summer. The 23-year-old, however, was keen to move this summer having previously been frustrated by Celtic demanding £20 million for the player.