The 33-year-old Francis was designated for assignment by the A's after posting a 6.08 ERA over 13 1/3 innings in nine relief appearances. Francis is expected to join the Yankees on Saturday in Baltimore.
"Obviously he's a guy that was stretched out, but he is not now," manager Joe Girardi said. "He's a guy that's probably in the 30-to-40-pitch range right now. We'll talk to him when he gets here and see what he's been doing the past few days."
Francis started the year with the Reds, starting one game -- giving up three runs on five hits in five innings -- before Oakland claimed him off waivers.
Francis, who won double-digit games for the Rockies in each season from 2005-07, has not started regularly since 2012. He split time between Colorado's bullpen and rotation in 2013 before he became a free agent.
Francis has a career record of 70-80 with a 4.95 ERA, and could be used to help out on Sunday in a start that was originally assigned to Masahiro Tanaka.
"I really haven't seen him pitch a lot," Girardi said. "I'll get to know him as soon as he gets here."
Bryan Hoch is a reporter for MLB.com. Follow him on Twitter @bryanhoch and read his MLBlog, Bombers Beat. This story was not subject to the approval of Major League Baseball or its clubs.