Pena, who is 34 and married, emerged from the rescue capsule Wednesday after being trapped underground with 32 other miners for 69 days.
Pena was reportedly among the most depressed of the trapped men and asked rescuers to send down a photo of the sun. He jogged regularly in the adjacent tunnels that weren't blocked by collapse.
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After hearing that Pena was an Elvis fan, Elvis Presley Enterprises sent various gifts to Chile, including a picture, DVDs, CDs, a book and sunglasses. "Viva Las Vegas" and "Jailhouse Rock," were among the movies.
"It just seemed it was the right thing to do for someone who is an Elvis fan, to send him some things that might keep them entertained," company spokesman Kevin Kern said.
After his rescue, an invitation was extended for an all-expenses-paid trip to Memphis and a special tour of Graceland, where Elvis lived for 20 years before he died on Aug. 16, 1977.
"We're going to make him feel like a rock star," Kern said.
The Graceland site includes his mansion, the graves of Elvis and family members, and various exhibits featuring The King of Rock 'N' Roll. Thousands of fans of his movies and music flock there each year from around the world to commemorate his death with a candlelight vigil.
Kern says he does not expect an answer from Pena to the invitation any time soon, saying there are more important things Pena needs to take care of.
"I'm sure they've got a lot of reuniting to do," Kern said.