Wiggins expressed an interest to leave ahead of the loan window closing last night after Championship outfit Watford had made approaches to sign him.
But despite tabling two firm bids for the 22-year-old, the Hornets were rebuffed, leaving Wiggins to see out at least the remainder of the season at Dean Court.
Watford's offers were believed to have been around the £150,000-mark, with Cherries valuing a player they signed for just £40,000 in the summer at more than double that figure.
Bradbury said: I've spoken to Rhoys about the situation and he is 100 per cent with us. He said he was right behind what we are all trying to achieve, which is great for me.
He is a great young player with massive potential and has a big future ahead of him. He is a big player for us and has been influential all season. To have let him go at this point would have been disappointing.
Wiggins, who has been troubled by a buttock injury, is expected to face a late fitness test ahead of Cherries' League One showdown with Charlton Athletic tomorrow (1pm kick-off).
Bradbury added: He is a professional footballer and, if he plays, everyone is looking at him. He will do a job and always gives 100 per cent. He won't accept anything less and neither will anyone around him. He has been great for us and long may that continue.
It is testament to him that Watford came in for him because he has done so well this season.
If the price had been fair for both parties, we wouldn't have stood in his way. The chairman has proved that with letting other lads go.
But, also, we don't want to become a club that sells players on the cheap every time somebody does well. We have got our own aspirations to be successful and have got a chance to do that this season.
Mitchell, who had the final say on rejecting the offers, said: We don't want people to come here and just be happy to play at this level. We want people to try to improve themselves all the time. If we have got 26 players doing that, it is the best thing for the club.
We want progressive players who want to work hard. If situations arise that other clubs want them as well, we have to deal with it. I wouldn't want Rhoys to be penalised for it. He just wants to do what is best for him and most people would be the same. I am pleased he is here and just want to forget all about it.
Mitchell, meanwhile, was keen to distance himself from suggestions from Wiggins that certain assurances had been made to the player when he signed for the club in the summer. Mitchell said he had sanctioned the move but not been involved in the negotiations.