Arsenal are "fighting" to keep wantaway centre-back Laurent Koscielny, head coach Unai Emery said in a news conference on Wednesday.
The Frenchman refused to go on the tour over doubts about the club's desire to keep him at the Emirates, sources told ESPN FC.
"I fight to convince him, but I respect his decision, a very personal decision, and that situation can only manage with one argument," Emery said. "I haven't lost my hope that he will be with us next year.
"He's a very important player for us. We are speaking with him and with the team."
Emery had previously said that he was waiting on the decision of Koscielny over his future at the club.
Arsenal signed French centre back William Saliba from Saint-Etienne for £27 million last month, but loaned him back to Saint Etiene on a season-long loan.
Emery continued: "We are speaking about the centre backs, but we have a lot of centre backs, we have good centre backs, and if we can sign one more centre back, it is because he is clearly better than we have. Because at the moment we have six."
Arsenal were quiet for much of the transfer window-- which resulted in fan unrest at the lack of investment from club owner Stan Kroenke -- but have since loaned Real Madrid midfielder Dani Ceballos, along with Saliba. Young Brazilian striker Gabriel Martinelli arrived earlier in the window.
The Gunners are expected to complete a five-year, €80m deal for Lille winger Nicolas Pepe on Tuesday, according to ESPN FC sources. Pepe was in London to complete a medical on Tuesday.
Emery said that the club expect to remain active in the transfer market this summer, saying, "we are working on the possibilities to sign players.
"We signed Ceballos, and we have a possibility with Pepe. We are looking for the possibility to improve, but only if we can take more steps in the squad.
"Our target is to sign another player if he can really, really improve us. Our target is continuing to sign another player if we can."