Cairney - told he could leave Leeds as a 16-year-old for being too small - was the standout player at Elland Road as he made his old club wish they had kept him.
But with Tom Huddlestone showing England form in Hull colours there does not appear to be an immediate future for him at the KC Stadium, and Tigers boss Steve Bruce indicated after his side's game at Liverpool that he expects a deal to be done soon.
"Steve Bruce must be sick of my number coming up on his phone," Bowyer said.
"Tom is a Hull player and we must be respectful of that but Steve has been great to allow Tom to come here with us. We have to see where it goes from there.
"He's a talented footballer that's for sure. The deal is work in progress and we will speak to Hull in the next couple of days."
With Cairney looking likely to become a permanent fixture at Ewood Park, Rovers fans will now be hoping the departures door stays bolted as far as Rhodes is concerned.
The Scotland international's goal was his 16th of another prolific season and Premier League clubs are almost certain to call, but Bowyer is sure any offers will be resisted.
"He increases his value with every goal but the owners don't want to sell him, it's as simple as that," he said.
"You can never say never at any price, look at Ronaldo or Gareth Bale with a monster, monster bid. But we are in no rush.
"I would expect a lot of interest but I would expect interest in a lot of players. Jordan has a good relationship with everyone at the club and wants to get to the Premier League with Blackburn."