Everton had been on the brink of appointing Vitor Pereira as boss at Goodison Park to succeed Rafael Benitez, but there was a fan backlash against the Portuguese.
The Toffees held further talks this week with their managerial candidates, including former Chelsea boss Lampard.
Lampard impressed Everton in his second interview and as a result has now been offered the job at Goodison Park.
If he accepts the post then he is expected to be in place next week.
Lampard will take over a side haunted by relegation worries following a prolonged period of poor form and will need to turn the ship around soon.
Everton are also looking to make signings before the transfer window slams shut on Monday night.
It remains to be seen if Lampard has suggested any players to the Everton board, with time now limited in the window.
