Rhodes, who is the Championship's second-highest goalscorer this season having found the net 16 times, has been linked with a host of Premier League clubs including Crystal Palace, Cardiff City and Hull City.
While Rovers' captain Dann is reported to have also attracted interest from Palace, but Bowyer maintains that he has the backing of the Ewood Park board.
"The owners want to reduce the wage bill but are very supportive of signings we are bringing in," he told the Telegraph.
"They are ambitious. They would love the club back in the Premier League. They understand a lot more now that it is going to take some time, step by step.
"They got some bad press when they first arrived. I can only take them as I find them.
"They've been very supportive of me, very positive.
"They want this club to succeed. They don't put pressure on me. They recognise the mistakes made in previous years."