It was Palmer, now a full England international and the star man at Stamford Bridge, who came out on top.
"A few players ended up taking their PlayStations," Archer explained to the Daily Echo, reflecting on the summer of 2023, which saw him become a European champion with the Young Lions.
"We had a lot of downtime, it was probably the longest four weeks of my life," he joked. "It was all worth it in the end.
"I remember playing Cole on FIFA about ten times. We were just going back and forth. He did get the better of me. He'll probably tell you I still owe him some money."
Cameron Archer (centre, holding shirt) and Cole Palmer (bottom right corner) won the U21 Euros with England in 2023.Cameron Archer (centre, holding shirt) and Cole Palmer (bottom right corner) won the U21 Euros with England in 2023. (Image: PA).Now the top goalscorer for their respective clubs in the Premier League, Archer is hoping for a different outcome when the two renew their rivalry.
Russell Martin's Southampton, who sit bottom of the Premier League table, welcome Enzo Maresca's Chelsea to St Mary's on Wednesday.
Much like Palmer does for the Blues, Archer could hold the key to unlocking three points for a depleted Saints side.
READ MORE: Crisis? Every Southampton player unavailable against Chelsea
The 22 year old has scored four goals across all competitions during his first few months on the south coast. Two of Archer's goals have come in the Premier League - but he should have scored more.
A would-be winner against Brighton was ruled out by a controversial VAR decision on Friday. He missed a penalty against Manchester United and struck the crossbar against Manchester City.
Cameron Archer had a goal ruled out against Brighton.Cameron Archer had a goal ruled out against Brighton. (Image: PA).Playing in the Premier League has always been the goal for Archer. The forward comes from a sporting family.
The youngest of four brothers, football was always on the agenda in the Archer household. His older brother, Jordan, was contracted to Bury when the Greater Manchester side were expelled from the EFL in 2019.
He subsequently joined various clubs before hanging up his boots after a knee injury. So did growing up in a football-mad family help Archer? "Yeah, definitely.
"When we were in the garden, I was the smallest and the youngest. I needed to make the most of my time on the ball. That helped with my upbringing."
Cameron Archer joined Saints from Aston Villa this summer. (Image: PA) Growing up in the Midlands, Archer had plenty of prestigious academies on his doorstep. He had trials with Wolves, Birmingham and West Brom before joining Aston Villa from Walsall at the age of eight.
He climbed the ranks at Villa before getting his first taste of senior football in the National League when he joined local side Solihull Moors on a season-long loan.
"It was a difficult season, especially with Covid as well," Archer explained, reflecting on his early career.
"I think I did good in my first couple of games but then the gaffer who brought me in got sacked.
"A new manager came in and I struggled to get back in the team. January came and the loan got extended. I was there for the rest of the season but it was difficult.
"I knew I had to go on loan because playing under-23s football is not good for you, really. I think, whether it went good or bad, the loan at Solihull Moors helped me."
Solihull Moors have agreed a loan extension that will see Cameron Archer remain with the club for the rest of the 2020/21 season. 🤩
💛💙 #SMFC #YourSolihullYourClub
— Solihull Moors FC (@SolihullMoors) January 4, 2021
Four goals is a decent return for a young player taking his first steps in the men's game, but things changed rapidly when he returned to Villa.
During pre-season, his under-23 teammates were invited to train with the first team in London. "I was the only one who stayed behind," Archer revealed.
"I trained with the under-18s then the first team needed me for an in-house game. I played the game and scored a hat-trick. That's where everything changed. I'd train with them all the time from there."
Having caught the eye of Villa boss Dean Smith, Archer was handed his full debut in a Carabao Cup second-round tie at Barrow.
"I was meant to be on the bench at first," he recalled. "The Brazilian striker, Wesley, was going to Club Brugge. I ended up starting and scored a hat-trick."
"In your head, it depends on the situation you're in at the moment. When you're that young, it's difficult to think too far ahead.
"As you get older, the belief is there. I think I just kept my head down and kept working hard. I always say, I just control the things I can control. Let everything else do its thing."
Subsequent successful loans with Preston and Middlesbrough gave Archer further exposure to senior football and aided his development.
He fired 18 goals in 40 games across both spells in the second tier before leaving Villa permanently to join newly promoted Sheffield United in 2023.
When the Blades were unable to avoid immediate relegation back to the Championship, Villa triggered the buyback clause inserted in Archer's contract.
Cameron Archer joined Saints from Aston Villa this summerCameron Archer joined Saints from Aston Villa this summer (Image: Stuart Martin) "It was something that Aston Villa wanted to put in my contract. It benefited me because if we didn't do well at Sheffield, I'd end up going back.
"I think the idea was if I went back I'd get sold again anyway. I think it benefited both parties."
Archer's journey brought him to Saints in the summer. A conversation with Martin convinced the youngster that this was where he needed to take his next steps.
Always willing to learn and improve, Archer has made a decent start to life at St Mary's but there is an understanding more will come.
"I've been up and down the last three or four years. I haven't really settled fully. But now I'm here at Southampton, I feel good. I feel like I can fully settle in now."