The Record claims that the Bhoys' latest offer - which could be lodged as early as today - could be around the £3m mark.
The Championship side had previously wanted as much as £6m for the former Chelsea man earlier in the window, according to the Daily Mail, but the Birmingham Mail recently claimed that they may be willing to do business for around £4m after the Scottish Premiership champions had made it clear that they were determined to land him, but less keen to pay the full amount.
And while Celtic's latest offer will still fall short of the Villains' valuation, it is perhaps possible to imagine that Roberto Di Matteo may be willing to allow a deal to go through.
Roberto Di Matteo
The continual speculation regarding his future is surely doing his squad no favours as they try and create a team capable of returning to the Premier League at the first time of asking.
And although he wants more money for the wide-man, he may perhaps be willing to consider a deal if Celtic do make the bid that the Record expects them to make as it gives him enough time to find a replacement, while holding out for the asking price may force him into a panic buy as the deadline looms.
Celtic manager Brendan Rodgers