Circe does not offer Odysseus immortality. She makes her appearance in Book 10, where she turns many of Odysseus' crew into pigs. One man escaped her treachery and went back to tell Odysseus of the situation, so Odysseus headed to Circe's palace. On his way there, he met Hermes, who gave him a way to avoid being put under Circe's spell and told him how to avoid Circe's other treacheries. Once he defeated Circe, Odysseus insisted that she turn his men back into men, and then they all stayed there for a year, living a life of luxury. When they left, Circe let them go peacefully and told Odysseus he needed to go to Hades to get directions from Tiresias.
However, the nymph Calypso does offer Odysseus immortality. In Book 5, Calypso is forced to allow Odysseus and his crew go free. When she talks to Odysseus and learns how eager he is to go home, she says:
"Son of Laertes, versatile Odysseus, after these years with me, you still desire your old home? Even so, I wish you well. If you could see it all, before you go--all the adversity you face at sea--you would stay here, and guard this house, and be immortal..." (Bk. 5, lines 212-218)
So it is not really an outright offer of immortality, simply a statement that if he were to stay with her on her island, he would be immortal.