"We have spoken to Dave Jones about the possibility of signing Jay," Jenkins told the Echo. "We were interested in a number of strikers and that was one conversation we had.
"He is not a player that is within our price range, though. Cardiff's valuation for Jay is miles out from where we put ours.
"That is where things stand at the moment. We couldn't come anywhere near Cardiff's valuation.
"It is not a disappointment, though. It's fine for us, no concern whatsoever."
And Jay's reaction.
"You can kill that story now - I'm not going there," he said.
City Chairman Peter Ridsdale called Swansea's approach a publicity stunt.
"I can only assume it's more for the publicity than a real interest," Cardiff chairman Peter Ridsdale told BBC Sport.
"I'd have thought it was as much tongue-in-cheek as anything else. If they were serious they would've rung me."
Bothroyd skippered City in the FA Cup win over Bristol City in midweek and will line up against Leciester later today.