The seed of Israel is the seed of Ephraim; the two names are used synonymously in scripture (eg., Isaiah 7, 9:9, 11:13, 28:1, Jeremiah 7:15, 31:20, etc.). Ephraim was Jacob's adopted half-Egyptian son whom he made his firstborn that inherited all of God's promises to Abraham Isaac and Jacob. Egypt symbolizes the world, which is significant, and the fact that Jacob's firstborn was adopted is very significant because it points to the adopted sons of God in Christ inheriting the promises.
This admixture of natural and foreign seed in the adopted son who inherited the promises foreshadowed God's removal of the difference between his people (circumcision) and the world (uncircumcision) and combining them into one new people of God, as it is written:
Wherefore remember, that ye being in time past Gentiles in the flesh, who are called uncircumcision by that which is called the circumcision in the flesh made by hands; That at that time ye were without Christ, being aliens from the commonwealth of Israel, and strangers from the covenants of promise, having no hope, and without God in the world: But now in Christ Jesus ye who sometimes were far off are made nigh by the blood of Christ. For he is our peace, who has made both one, and has broken down the middle wall of partition between us; Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of two, one new man, so making peace; Ephesians 2:11-15
Under the old covenant God's people were determined by circumcision, not by ethnicity or race, because anyone in the world could become a son of Israel by becoming circumcised and keeping the law. This foreshadowed the eternal reality of anyone in the world being able to become a son of God by becoming spiritually circumcised through faith in Christ.
Jacob prophesied that the seed of Ephraim (Israel) would become a fullness of gentiles.
And [Jacob] his father refused, and said, I know, my son, I know. He [Manasseh] also shall become a people, and he also shall be great: but truly his younger brother [Ephraim] shall be greater than he, and his seed shall become a fullness of gentiles. Genesis 48:19
The natural nation of Ephraim/Israel was sown in death among the nations in 722 BC, and raised out of the dead on the day of Pentecost when God's holy nation consisting of Abraham's natural seed and the seed of foreigners was born from above by the holy spirit on Mt Zion.
During the festival of Pentecost, the high priest would wave two loaves of leavened bread before the God. The two loaves symbolize the circumcision and the uncircumcision combined into one people of God, and bread with leaven symbolizes sin (unlike unleavened bread that symbolized the sinless Christ), which the body of Christ has until it is perfected
When the prophets prophesied about the restoration of Israel, this is what they saw in spirit. An eternal, holy nation consisting of Abraham's spiritual seed. And true to Jacob's prediction, Ephraim became greater than his brother by becoming a fullness of gentiles.