Davis, who sits on the board of BP, will take over from Robertson after the company's annual meeting in May.
Sky News reported that Davis, a former McKinsey partner, had been in job talks with Rolls Royce for several months.
Davis is well known to Rolls Royce board member Iain Conn, who led the search for a successor to Sir Simon Robertson, the daily reported.
Conn also head BP's refining and marketing operation.
Davis' appointment is expected along with the British aerospace and defence group's reporting its full-year results on Thursday.