Jurgen Klopp’s side are understood to have held talks with the winger’s agent but the Gunners have stunned everyone by beating off some stiff competition for one of the most in-demand players in world football.
Liverpool are already well-stocked in that area so missing out is not a huge blow by any means.
But they are beginning to look a little light out wide in terms of competition.
Mohamed Salah and Sadio Mane are the most prolific wingers in the division but both are already in a race against time to be for the start of the new campaign because of their involvement in the Africa Cup of Nations
Xherdan Shaqiri remains a useful back-up option but Divock Origi can cover out wide if needed while Rhian Brewster remains a relative unknown and Harry Wilson looks set to leave before the close of the window.
So what now for Liverpool after missing out on Pepe? Do they need to go back into the transfer market?
Klopp can hardly afford to rest on his laurels if they want to seriously challenge Manchester City for the title again.
Here, Express Sport explores three wingers they could potentially go for between now and August 8.
Philippe Coutinho (Barcelona)
Linked with a return to Anfield for almost the entire window, it seems unlikely – yet not impossible – that Coutinho will be back on Merseyside when the season starts.
Barcelona need to sell after spending a combined £170million on Antoine Griezmann and Frenkie de Jong, and they are still trying to bring Neymar back from Paris Saint-Germain.
Coutinho, 27, is already facing a season on the fringes and Neymar’s arrival would restrict his first-team opportunities even further.
Whether Liverpool fans would welcome him back with open arms remains to be seen as there is still some bad blood after he forced his move to the Nou Camp last January.
The Brazilian would not be an automatic pick for Klopp if he were to come back either but he would add versatility to his prolific forward line as he could play behind the striker or on the left when Mane is unavailable or rested.
Paulo Dybala (Juventus)
While not the out-and-out winger Liverpool might need, Dybala is one of very few genuinely world-class attackers who appear to be available this summer.
The 25-year-old struggled baldy for form last season after Cristiano Ronaldo joined the club but make no mistake: Dybala is a player of the very highest order.
Making a move for the Argentina star would not only provide a huge boost to their own hopes for the season but deal a bitter blow to rivals Manchester United and Tottenham who are also keen to sign him.
United have already opened a dialogue with Juve and are currently big favourites to sign him but he might prefer a move to Liverpool with the carrot of Champions League football and a place in the Club World Cup on offer.
They will probably need to act fast, though, and it appears to be a long shot as things stand.
Wilfried Zaha (Crystal Palace)
Zaha is desperate for another crack at the Champions League following his failed move to Manchester United in 2013.
He looked certain to leave earlier in the window with Arsenal and Tottenham both keen to take him to north London but they have now concentrated their efforts on alternative targets and Everton are the only club in the race to sign him.
Everton have already submitted a bid that was rejected and they are expected to go back with improved terms on the table before the deadline.
At around £60million, the 26-year-old looks a little bit overpriced but
Liverpool could do a lot worse than taking a look themselves if they fell they really need a new winger.