Glasgow Rangers, as well as several other sides from the Championship, are looking to sign the Welshman.
The Express and Star reports that the Saddlers want around £1m for the highly-rated 23-year-old, but would also be willing to accept an offer of around £750,000 if add-ons bolster the overall fee.
However, the Championship side's bid appears to have been well short of that, with the Express and Star suggesting that they offered £250,000 for the forward, which was also less than the £400,000 bid that Barnsley had knocked back previously.
Although the League One side would have probably given the Millers' offer little thought given how short of the asking price it fell, they may now be worried that they may be about to be flooded with bids.
The Express and Star reports that Scottish Premiership side Rangers are monitoring the former Shrewsbury man's situation, while Rotherham's Championship rivals Nottingham Forest and Reading are also keeping tabs on the striker.
It is hardly surprising that Bradshaw is attracting such attention. He has scored 17 league goals in each of the last two seasons, with his strikes this term helping his side qualify for the play-offs.
But with Jon Whitney's men losing their semi-final, they may find it difficult to keep the forward at the club with teams from higher divisions looking to lure him away.