The former Everton boss was widely expected to take some of his Goodison Park backroom staff with him to Old Trafford, particularly Lumsden who worked with him at Preston 11 years ago.
Manchester United are expected to announce the departure of Sir Alex Ferguson’s assistant Mike Phelan and goalkeeping coach Eric Steele, although the Scotsman’s first-team coach Rene Muelensteen could yet stay.
Everton keeper coach Chris Woods may also be considering his future amid the break-up of the Toffees’ tightly-knit coaching staff as the club prepare to usher in a new era with the appointment of a new manager.
Although the new manager is likely to bring in his own coaching team, Everton are set to retain the structure in place above with respect to player recruitment.
Round, 42, joined Everton from Derby County in 2008 after a short spell working under Sam Allardyce at Newcastle United.