The 23-year-old has one year left on contract and is valued at £750,000 by the Saddlers, with the forward also interesting Preston North End and Birmingham City, with Rotherham United also linked.
Barnsley - who have signed George Moncur and Elliott Lee - have seen an opening bid of £250,000 rejected, but have now come back with a revised offer.
Wales international Bradshaw fired 20 goals for the Saddlers last term.
Rotherham will complete the signing of ex-Walsall winger Anthony Forde today, but will be priced out of any move for Millwall’s Sheffield-born forward Lee Gregory.
Burnley are set to step up their interest in Sheffield Wednesday defender Tom Lees, regardless of where Michael Keane stays at the club or not.
Ex-Leeds centre-half Lees emerged as a target for the Clarets amid talks that Keane could move to champions Leicester.
But the Foxes saw an initial £10m move turn down, although they are expected to make a renewed bid.
Discounting Keane’s fate, Burnley will maintain their interest in Lees, also being monitored by other clubs including Aston Villa, Newcastle United and Norwich City.
Sheffield United remain confident of securing the services of Kyle McFadzean, but face competition from Charlton Athletic for fellow centre-back and ex-Blades and Leeds defender Matt Kilgallon.
Hull City are in the market along with Sheffield Wednesday, Norwich City and Celtic for Sunderland striker Steven Fletcher.
Tigers boss Steve Bruce, yet to make a major summer signing, tried to sign the Scotland international on the last time that Hull were in the Premier League, but the striker stayed put on Weaside.
But Fletcher is now available on a free transfer, with several clubs circling.
Aston Villa’s Brad Guzan is the latest goalkeeping name linked with Middlesbrough, who have held talks about signing the 31-year-old.
Boro are in the market for two new keepers, with Tim Krul and Victor Valdes among other names mentioned.
Meanwhile, young Fiorentina forward Khouma Babacar is also on the radar of Boro, who are competing with Sassulo to sign the 23-year-old.
Despite starting last campaign strongly for La Viola, the Senegalese striker soon fell out of favour with coach Paulo Sousa and the club look set to cash in.
Leeds United striker Mirco Antenucci has joined newly promoted Serie B side SPAL in his native Italy despite late interest from Reading.
Antenucci spent two years with Leeds after signing from Italian outfit Ternana in the summer of 2014.