Everton return to pre-season training on Thursday and manager David Moyes, away on a family holiday until then, would like Round installed as quickly as possible.
But there is a clear dispute between the clubs over when Round's contract at St James' Park officially ends.
Everton believe Round's contract ends on Wednesday and that he would be free to take up his appointment as Moyes' number two.
But Newcastle insist that they hold a one-year option on the 37-year-old former Derby player, and are demanding compensation for the coach who was former Magpies manager Sam Allardyce's first appointment this time last summer.
An Everton spokesman said: "There is an issue over an option on the contract and the clubs are in ongoing discussions over a solution.
"We believe it is something that can be sorted out to allow the appointment to be made."
Everton were given permission to speak to Round last week but cannot take the appointment further until the compensation wrangle is concluded.