"When I took Spanish in fourth grade, my name was Elena. They gave me the name -- 'me llamo Elena' [my name is Elena] -- because I couldn't roll my r's very well," she told the Las Vegas Culinary Workers Union on Tuesday.
The controversy comes only days before Saturday's caucuses in Nevada.
Some on Twitter suggested she was engaging in a form of pandering.
"Oh please, [...] The identity pandering is shameless," said Mark Levin, radio host of Fox News Channel's "Life Liberty & Levin."
Conservative author Ben Shapiro tweeted: "She also once watched an episode of 'Dora the Explorer.'"
Others, including Daily Kos writer Gabe Ortiz, piled on to the Twitter criticism. "I don't know why I'm the one feeling embarrassed but I am," he said.
Klobuchar also caught attention for previously being unable to name the Mexican president while appearing on Telemundo last week.
On Tuesday, Klobuchar defended herself but reportedly mispronounced the president's name.
“Well, first of all, I would like to give my greetings to President Andrés Manuel López Obrador,” she said.
Klobuchar had said she had a busy day in the Senate, regarding not knowing the president's name.
“When that happened, for what it’s worth, I had been in the Senate all day, we had six votes, including a resolution to be a check on the president so he doesn’t go pell-mell into war with Iran,” she said.
She also rebuffed criticism from former Indiana Mayor Pete Buttigieg, saying: "This isn’t like a game of 'Jeopardy!' This is about, to me, experience, and I have so much respect for him and his experience, but my experience is different."