The Gunners have already confirmed a deal to sign Sead Kolasinac when the window opens on July 1, although the club are thought to have been unsuccessful in an ambitious £87m bid for AS Monaco winger Kylian Mbappe.
Wenger has previously suggested that he will bring no more than three new faces into the Emirates Stadium this summer, and he hopes to wrap up all of his business by the time his side travel to Australia for a pre-season tour on July 13.
"Between now and the tour we will work every day to get the best possible environment around the team and to strengthen the team," Wenger told the official website of the club's summer tour.
"That's basically a lot of dark, invisible work. But that is demanded and that means never leave your seat without your phone, never go to bed without your phone, because anything can happen. We will basically be on alert 24 hours a day."