Bassitt expects that time to come before season's end, and Melvin isn't ruling out a potential start for the right-hander during the club's next homestand, which begins Sept. 22.
"I don't foresee him pitching in a game on this road trip, but potentially when we get home," Melvin said. "Ultimately we'd like to get him back to start again, as opposed to just getting him in there pitching out of the bullpen."
Bassitt, who returns to Chicago for the first time since being traded by the White Sox in the Jeff Samardzija deal, is 1-6 in 10 starts for the A's despite posting a 2.73 ERA.
• Outfielder Billy Burns (right hamstring) was "feeling a lot better" Sunday, according to Melvin. Burns will run the bases in Chicago on Monday and potentially be available as soon as Tuesday.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.