BBC Sport report on Saturday that the transfer, paid in instalments, will be completed within the “next 24/48 hours”.
The 24-year-old had also been in talks with Napoli but now looks set for a move to the Emirates after his agent rejected the Italian club.
He is expected to sign a five-year contract at Arsenal.
News of Pepe’s imminent arrival comes after the Gunners completed the loan signing of midfielder Dani Ceballos from Real Madrid.
The one-year deal reportedly cost Arsenal £15m and does not include a compulsory purchase clause for next summer.
Ceballos, who has struggled for regular game time at the Bernabeu, hopes to improve his chances of a place in Spain’s Euro 2020 squad with a season in the Premier League.
The 22-year-old is Arsenal’s third signing of the summer, following the arrival of teenage striker Gabriel Martinelli and St Etienne youngster William Saliba, who has immediately been sent back on loan to the Ligue 1 club.
“Saliba was different, part of the deal is for him to be on loan for one year with his club,” said Arsenal chief Raul Sanllehi. “We are pleased about that because the level of intensity and the level of demand in the Premier League is very strong.
“He’s probably too young for that yet, as I said, we need to protect our young talent. We think it will be good for him to be in a less demanding league to continue his formative period, so we’ll have a top defender for a decade.”
Elsewhere, the Gunners continue to be frustrated in their pursuit of Celtic full back Kieran Tierney.
The Scottish champions have rejected two bids for the youngster and are demanding Arsenal meet their £25m valuation.
And manager Neil Lennon has insisted nothing has changed since the last rejected offer. “It’s as you were a week ago, two bids have come in and two bids have been rejected,” he said. “That is where we are.”
Arsenal are also in danger of losing one of their most highly-rated young prospects: 18-year-old midfielder Xavier Amaechi.
The teenager is moving closer to an exit, with a switch to Germany the most likely outcome. Talks have been held over a new contract, but Amaechi, who has just one year remaining on his current deal, is expected to leave.
Hamburg are one of the German clubs believed to be interested and, according to Bild, have seen a bid of €2.5m accepted.
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