And Everton midfield player Mo Besic has returned to Championship club Middlesbrough on a season-long loan deal, the Teessiders have announced.
Barnsley striker Bradshaw will undergo a medical later today, with the player expressing a wish to leave due to his personal situation with his partner living in the south.
Reports suggest that the Reds have agreed a £1m fee with Millwall, rising to £1.25m.
Winkler said: “We are very disappointed. Everybody; the staff and players.
“He is a really good player who fits into the squad and style of football and has personality and character. He’s a really nice guy.”
On the deal, he said: “We are all on the same page, the board and the manager. We were informed on Tuesday about the situation.
“It is not easy (for Bradshaw). It is a little bit complicated and a private situation. He wants to move to get close to the south. His girlfriend lives in the south and they want to raise a family. We understand, but from a football point of view, it’s disappointing.”
On a potential replacement, he added: “We are watching lots of players. Hopefully, we will sign a replacement, but we are not sure. We want to find the right replacement.”
The 25-year-old Besic, who made 15 appearances during the second half of last season as Boro reached the play-off semi-finals, completed his temporary move today as manager Tony Pulis finally got his man after lengthy negotiations.
A statement published on mfc.co.uk read: “The club are delighted to confirm the signing of Bosnia international Mo Besic from Everton on a season-long loan.
“Tony Pulis has made no secret of his admiration for the player and is delighted to secure his signature for the duration of the current season.”
Boro had hoped to push through a permanent transfer ahead of the August 9 deadline but, having failed in that mission, continued their pursuit amid competition from a series of rivals and have now got their reward.
Pulis had also hoped to persuade Besic’s team-mate, Yannick Bolasie, to head for the Riverside Stadium on loan, but he now believes the 29-year-old is to join Aston Villa instead.
Bolasie’s decision looks set to mean Boro winger Marvin Johnson will not be going to either Sheffield United or Hull.
The Blades’ move for Villa’s Scott Hogan could depend on Steve Bruce’s side signing Shane Long on loan from Southampton
Turkish club Galatasaray have joined the chase for Hull City’s Poland World Cup winger Kamil Grosicki.
Leeds United have agreed a deal which will see midfielder Oriol Rey join Spanish side UB Conquense.
Steve McClaren is hoping to sign two new strikers at QPR - Brighton’s Tomar Hemed and Burnley’s former Huddersfield Town player Nahki Wells.
Brighton and Hove Albion still working on arranging a season-long loan for Richie Towell to return to Rotherham.