The left-back yesterday formally declared his desire to leave Dean Court.
Bradbury revealed the club had turned down Wiggins's request and that he was keen to retain the services of the former Welsh youth international, who joined Cherries on a two-year contract in July.
Wiggins's decision came after Cherries declined bids from Watford for the Londoner. The offers were described as miles away from our valuation by chairman Eddie Mitchell.
With the loan window due to close at 5pm today, the Hornets could come back in for Wiggins ahead of the deadline, with a view to making the deal permanent in the summer.
But Bradbury told the Echo: We are trying to keep the lads together, build the club and add additions to it that are going to be positive for us. We don't want to be selling any of our players.
Rhoys has been influential this season and different class for us. To sell him would be very negative from our point of view.
For him, it is attractive because it is a Championship club that is going places and it is close to home for him.
But from our point of view, we want to keep him. He is a valuable asset for us and a great player with a great future.
I want him to knuckle down and put in 100 per cent for the rest of the season and see where it takes us. We have got a job to finish and that is down to every one of us.
Chairman Mitchell reported that Cherries had received two firm offers from the Hornets but said they were a long way off what the Dorset club would consider as an acceptable figure.
He insisted he was determined to retain the services of Cherries' leading stars.
Mitchell added: Our first aim is to go up this season and until we are out of that contention, we don't want to sell anybody.
If somebody comes in with what we would class as a very, very good offer for somebody, then we have got to look to see whether with that money we can replace the player we are losing and have surplus.
At this moment in time, Watford are miles away from it, so it is not even a consideration for us.