The Toffees are currently looking for their next manager following the sacking of Roberto Martinez last month.
A number of names have been linked with the position, but developments this week have seen the Southampton manager become the hot favourite for the position.
The Blues have reportedly pushed hard to secure his signature, and it appears their efforts have been successful, with the Daily Mail reporting that the Dutchman has agreed terms with the club.
According to them, the Saints boss has been swayed by the transfer funds that have been pledged by Farhad Moshiri after meeting with the Everton shareholder.
He had initially been reluctant to leave St Mary’s, and indeed at one stage appeared to be close to a new deal, but the financial restraints at the club have convinced that a move to Goodison Park would be for the best.
A deal is not completed yet as Everton will have to pay up the final year of his contract with Southampton, which is understood to come to around £3.5million.
It is believed that Koeman is keen for it to be finalised before he goes on holiday next week as he plans to start work on Merseyside the following week.