"But my mom doesn't have any papers," the girl said.
"We have to work on that, we have to fix that, in that everybody's got to work together in Congress to make sure that happens," the first lady said reassuringly.
Since that exchange, the seven-year-old's innocent face has been everywhere here as the Lima press retraces her family's journey, full of hope and sacrifice, from the working class suburb of San Juan de Lurigancho to the promise land up north.
Her parents Natalia and Angel set out eight years ago, leaving behind a year-old baby, July, to be raised by her grandparents.
"They went to work over there because there was nothing to do here," said the girl's grandfather, Genaro Julca.
Angel found work as a carpenter, Natalia as a domestic worker, and they send money to Peru every month.
"My daughter has an honorable, hard-working home," said Julca, who only wants the United States "to allow the young daughters to be reunited."
July and Daisy, who was born in the United States, have only seen each other once, except through webcam visits.
"I hope that they give my mother papers, so that she will be calm and no longer afraid," July said on Peruvian television.
Since her daughter's sudden fame, Natalia has stayed out of sight.
While she has not been bothered, she has followed a lawyer's advice not to speak to the media, said Julca, who says he "trembles with fear" that his daughter might be deported.
Instead, politicians have been speaking up for her, taking up her cause at a time when immigration reform is a hot topic across the United States, especially after a controversial Arizona law that allows police to question anyone they suspect does not have papers.
The Arizona law has drawn protests from Mexican President Felipe Calderon that resonate throughout Latin America, home to the vast majority of the estimated 10.8 million immigrants in the United States illegally.
About one million Peruvians, both those with papers and those without, live in the United States.
Peru's President Alan Garcia has been quick to embrace Daisy's case, proclaiming his "pride that a Pervuian child symbolizes the immense problem of Latin American immigration in the United States."
"It's a problem unfortunately that is not on a path of a solution, but which arouses irrational responses like that of Arizona," he said.
Garcia, who is supposed to visit Washington in early June at Obama's invitation, added that "of course," he will express his concerns about immigration with the US president.
Daisy "is an unappointed, but effective ambassador" on an issue "so important for all Latin American countries," said Foreign Minister Jose Antonio Belaunde, underscoring Peru's intention to capitalize on the attention she has attracted.