The two north London clubs are understood to have joined the race to sign Karamoh with interest thought to be high from other Premier League clubs including West Ham, Newcastle, Everton and Southampton.
The French club are understood to want £9million for the youngster with Sky Sports Italia claiming that Inter Milan have already agreed to pay an initial £7million.
However, the Daily Mirror now reports that the Gunners and Spurs will enter the battle and could make a move before the new campaign begins.
Karamoh, who has one year left on his contract at Stade Michel D’Ornano, said recently: “If I had the choice, I would go to a club where I would play.
“I do not want to go to Monaco to go on loan. When I see (Allan) Saint-Maximin, who has been on loan at Bastia, I think, ‘I just played 35 matches this season,’ and I wonder if it’d be worth it.
“I want to play. I want to go to a club where I could continue to evolve to mature in two or three years.”
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Karamoh was born in Abidjan in the Ivory Coast before joining Paris-based club Racing Colombes 92 as a youth player in 2006 at the age of eight.
On December 7, 2015 he signed his first professional deal with Caen after moving there as a youth player in 2011.
After impressing for Caen II with 11 goals in 28 appearances he earned his chance in the first team, making his Ligue 1 debut in August 2016 as Caen beat Lorient 3-2.
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He only had to wait six days for his first start at the club, playing 90 minutes in a 2-0 away defeat to French giants Lyon.
The 19-year-old has really come to the fore this season for Caen with five goals and four assists in just 25 Ligue 1 matches from the right wing.
That form has attracted the interest of suitors all over Europe and his ability to play as a striker has led to comparisons with fellow Frenchman Kylian Mbappe, who took Ligue 1 by storm last term.
Karamoh, who could also represent the Ivory Coast, has been capped for France at U16, U17, U18, U19 and U21 level, scoring at every age group.
Only the post denied us a ludicrous goal.
A throwback to Moussa at his best.
Who would win between Arsenal 04 and Man City 18?
Jose is pretty passionate in his response.
The most joyous player in the Premier League.
There’s a total of 20 to get.
Including a couple of notable sons.
You’ll be hearing a lot more about Allison.
Of course he meant it. He meant everything.
The second half of an epic challenge.