A switch could potentially be done inside just 24 hours with a move now agreed.
Unai Emery’s side have acted quickly to beat their European rivals to the 22-year-old.
Liverpool and Manchester United have also been keen on Pepe this summer.
Meanwhile, Napoli were believed to be frontrunners up until Arsenal lodged their offer.
The Gunners have ultimately blown the Serie A giants out the water in the race to sign Pepe.
Napoli also met Lille’s price demands but were turned down by the player’s representatives.
With Napoli’s proposal rejected by Pepe’s camp, it leaves Arsenal free to complete a deal.
All that is left to sort is final details between Arsenal and Pepe’s representatives.
“Pepe is subsequently set to be an Arsenal player by Monday”
But it is not anticipated there will be any delays to the move as BBC Sport say a transfer is expected in the next 24 to 48 hours.
Pepe is subsequently set to be an Arsenal player by Monday.
It is unclear, though, when he would make his debut for the Gunners.
The Ivory Coast forward has had a delayed pre-season as he was international duty at the Africa Cup of Nations until July 11.
On Wednesday evening, Arsenal play away to French outfit Angers, who Pepe would have faced last season with Lille.
That is expected to come too soon for the versatile attacker, who primarily plays on the right wing but can also be deployed up top in a front two.
Arsenal, meanwhile, play Barcelona at the Nou Camp next Sunday before kicking off their Premier League season at Newcastle on August 11.
He could feature against La Liga’s reigning champions in a bid to get him fully ready for the game at St James’ Park.
Before all that, Arsenal host Lille’s Ligue 1 rivals Lyon tomorrow in the Emirates Cup.