When he was offered immortality if he would marry Calypso, he himself gave it up to go back with his wife Penelope.
She offers Odysseus immortality.
Odysseus is offered immortality and eternal youth by the goddess Calypso if he would stay with her, but Odysseus is drawn by his faithfulness to his wife to refuse Calypso. Odysseus strives to attain immortality, not by living forever, but by gaining a reputation that will have bards telling of his achievements. In this since, Odysseus achieves total immortality. In the since that he lives forever, alas, Odysseus is out of luck. Apparently between the giant cyclops and the feasting he forgot to look for the fountain of youth.
Calypso offers immortality to Odysseus
Calypso
immortality.
immortality
Immortality
Immortality
Immortality/Eternal Life.
Calypso offers Odysseus immortality and eternal youth if he would stay with her forever.
Odysseus' reaction is to enter the cave to see if anyone is home.
In book 5 of the Odyssey, Calypso offers Odysseus immortality and eternal youth if he agrees to stay with her on her island forever. She promises him a life of luxury and pleasure if he chooses to remain with her as her immortal consort.