"I named him after Derek Jeter because he's the team captain," Robert Villacis, the boy's father, told the New York Daily News. "He's going to grow strong like him, too."
The Daily News ran a story on Thursday with a photo of the boy -- decked out in a Yankees short and cap and a ventilator under his nose -- and the headline, "Meet another New York hero named Jeter!"
Derek Jeter wasn't aware of the story until television reporters informed him of it following Thursday's workout at Yankee Stadium.
"I did not know that. This is news to me," said Jeter. "The encouragement I would give him is to stay positive. That's the biggest thing. Any time you deal with adversity, you try to stay positive. I'm glad to hear he's doing well."
Jeter Villacis had a nine-hour surgery last week and his prognosis is encouraging.
"I was afraid I was going to lose him," Vanessa Villacis, Jeter's mother, told the Daily News.
Derek Jeter still remembers the intersection that baseball and life can have from the days following Sept. 11, 2001, when terrorist attacks killed thousands.
"It's kind of hard to imagine," Jeter said. "When it really hit home was after Sept. 11, we had an opportunity to go around and see how all the fans rallied around us. We feel as though we're playing a game, but it's fun to know that you have people who gain inspiration from you."
Did the captain have a message for Jeter Villacis?
"Keep battling," Derek Jeter said. "That's first and foremost. And always stay positive. That's what we try to do here. And I think you need to do that in everyday life."
Derek Jeter did get a chuckle that someone named their child after him. After all, Jeter isn't exactly a common first name.
"I hope the kids aren't too hard on him in school," quipped Jeter. "It's kind of hard to believe."