The Apostles' Creed uses everlasting:
I believe in the Holy Spirit, the holy *catholic church,
the communion of saints,
the forgiveness of sins,
the resurrection of the body,
and life everlasting.
Here is an article that captures the idea of everlasting life, that is something that we hope for. A promise of future reward.
The Gospels talk a lot about eternal life. See, e.g., Story of the Rich Young Man, John 3:16.
Which meaning for eternal are the Gospels using? May it be that eternal life is timeless life? What we are seeking, may be finding the life that is now and was and will be. Something like the perfect or ideal life.