Media playback is not supported on this device
The Gunners' offer - reported to be £6m - was rejected by Wanderers and criticised by their boss Owen Coyle.
In addition, Trotters chairman Phil Gartside re-posted a message on Twitter from a Bolton fan angered by the bid.
"That number is completely wrong," said Wenger. "If Gartside can say I am lying I am ready to confront him."
The Frenchman added: "I believe every negotiation is between two parties and you only have to sell the player if you are all right with the price.
"In this case the information is below what has been spoken about.
"You can believe Gartside or you can believe me. It's not right. It's not the truth.
"The truth is that we never speak about our negotiations.
"If a club doesn't want to sell a player it can keep him. When we sign a player we come out here and tell you and keep the price a secret.
"They do not need to sell the player, they can keep him. If you ask do I want to buy your house and you are not happy with the price you say no, that's it.
"Why should you feel insulted? I don't understand. If the player is not for sale they can keep him. I don't understand the problem.
"If we want to buy the player we buy the player. I do not have to explain for how much."
Media playback is not supported on this device
On Friday, in response to Arsenal's bid for England international Cahill, Bolton manager Coyle said: "When I make an offer for a player, I'd like to think it's in the right ball park.
"But this wasn't even close. I'm only passing it on as a third party, but the word derisory doesn't even cover it."
In June Coyle said he was resigned to losing the 25-year-old centre-back - who joined them from Aston Villa and is now in the final year of his contract - this summer.
Arsenal have a raft of injury problems, and defenders Laurent Koscielny and Sebastien Squillaci both missed the club's Champions League qualifying win at Udinese on Wednesday.
The transfer window closes on 31 August.
Arsenal have received nearly £60m for the sales of Cesc Fabregas and Samir Nasri this summer, and manager Arsene Wenger insists the money that has come in is available for him to buy new players.
Arsenal manager Arsene Wenger has reacted angrily to suggestions from Bolton that his bid for their defender Gary Cahill was "derisory".