The pair went to war during a ding-dong three-round scrap as part of the Sky Sports Boxxer Series in Liverpool on Saturday with Appleyard, who is eight years older than Kinsiona at 33, coming out on top via split decision.
Parson Cross-based Kionsiona had got under his opponent’s skin in the build up to the clash, which led to fireworks from the first bell on Bonfire weekend.
"Even before I jumped out the ring my manager Dave Allen got a message from Eddie Hearn saying he wants to put the fight on over 10 rounds,” said Kinsiona.
"The promoters at Sky Sports also want to put the fight on.
"I know it was the wrong decision and want to rectify that. Everyone saw I had the superior skill set.”
Kinsiona usually boxes at lightweight but would be willing to contest a rematch at the weight above, which Saturday’s contest was fought at.
Earlier this year he turned down a fight against Appleyard at super lightweight for the Central Area title.
Appleyard said: "I’m happy to have the rematch, it’s got to make financial sense and there’s got to be something on the line for it.
“I’m not fighting for peanuts so there needs to be a title on the line and good money. I’m not interested in the Central Area title, I’m looking for bigger things.”
The pair did patch things up following the final bell after Appleyard, who will not fight again this year, revealed Kinsiona apologised.
He added: “It felt personal for me, which was probably a bad thing for Levi in that he created that needle in a three-round format. I needed that without any sparring.
"What I found funny is before we went into this he was saying ‘this format suits me’ then in his post-fight interview he said it suited me more.
"Now he’s shouting for a championship fight when he’s never done the distance before.”
Kinsiona admitted he should have started faster in round one but said: “My stock has risen so much from this weekend and I think it has risen more than it would have if I had got the win because of the controversial decision.”