Maupay opened the scoring in Brentford's trip to Tottenham, which ultimately finished as a 3-2 victory for the hosts, on Wednesday night.
After finding the back of the net, Maupay mocked Maddison by copying his darts throw celebration, with team-mate Ivan Toney joining in on the act.
The pair shared a heated exchange after the celebration, though Maddison and Tottenham ultimately would have the last laugh by taking three points.
Neal Maupay (right) and Ivan Toney mocked James Maddison's dart throw celebration after the former opened the scoring during Brentford's 3-2 defeat at Tottenham
James Maddison later hit back at the Bees star with a social media post after the game
It's All Kicking Off co-host Chris Sutton claimed that Maupay 'couldn't lace Maddison's boots'
'Neal Maupay, the Brentford striker, opened the scoring for Brentford and chose to mimic James Maddison's dart throwing celebration. That went down badly with James Maddison. All got a bit ugly, bit of pushing and shoving.
'It was still going at the end of the game. A bit of aggro at the end of the game. Both managers asked about it. They both looked grumpy as hell about it. What do you make of all that?'
Sutton responded: 'Honestly, Neal Maupay couldn't lace James Maddison's boots. What is he doing? Honestly, what an idiot.
'Everybody knows that Maddison is 10 times the player that Neal Maupay is. Why he did it, I don't know. You can say it's a bit mischievous. You know, describe it as you want, but it was foolish. I mean, Maddison is Waitrose, Maupay is Lidl'.
After the game, the Brentford striker hit back on Instagram writing: 'Gutted we couldn't get the win. More goals and less relegations in my career than James Maddison. We go again Monday Bees.'
The Tottenham star posted a picture of Maupay on his Instagram doing the celebration, alongside the caption: 'A short story... COYS'.
Maupay and Maddison had an awkward exchange after the former mocked the latter's celebration
Maupay in retaliation said he had 'more goals and less relegations' than Maddison
Ladyman continued: 'What grabbed my attention was Maupay's Instagram feed that I spotted just as I was going to bed last night.
'So Maupay has gone on Instagram, he's posted a photograph of himself doing the Dart celebration. Got to understand that they've lost this game.
'He's posted a photograph of the darts celebration. And then he said, 'gutted we couldn't get the win. I've got more goals and less relegations in my career than James Maddison'.
'Well, first things first, Neal, you're a centre forward. James Maddison is not a centre forward. So I think it'd be deeply embarrassing if you weren't to have more goals than him.
'This is a guy, yeah, he's just scored four in four. He scored four in the bounce Maupay for Brentford.
'Previous to that, he was playing as though he was wearing concrete wellies. I mean, now he's behaving on Instagram as if he's got a record that George Best would have been proud of. I just find it a little bit peculiar.'
Ian Ladyman also highlighted Maupay's recent struggles in front of goal for Brentford
Sutton then interjected again to say: 'No. But do you know the most serious thing about it all? Brentford are in a relegation battle.
'And that's the biggest thing. And if I'm his manager, I'm absolutely furious that he's bothering getting in petty spats on social media when the most important thing is for him to find his form at Brentford.
'It doesn't look like his eye is on the prize and his head is in the right place at this moment in time and that would be my worry.
'He may well have a relegation on his hands this season so he needs to pull his finger out.'
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