The comments from Republican Rep Betty Brown came on Tuesday, when the state legislature heard testimony from a Chinese-American group on voting difficulties. Asian voters' names are often spelled differently on different documents.
"Rather than everyone here having to learn Chinese," Mrs Brown said to a representative from the group, "do you think that it would behoove you and your citizens to adopt a name that we could deal with more readily here?"
But as a New York City councilman demanded an apology, a spokesman for Mrs Brown defended her, saying she was not making a racially-motivated comment, but simply trying to resolve a problem of voter identification.
Mrs Brown said on Thursday that she had reached out to Ramey Ko, a representative of the Organisation of Chinese Americans.
"I tried to call him this morning to talk to him about it. And if I had offended him to apologise but I haven't received a call back," she said.
Mrs Brown told Dallas-Fort Worth television station KDFW that she "misunderstood a little bit about what he was presenting as the problem".
She also said "At 1 am, after we'd been through about 10 hours, I probably wasn't speaking just as I should."
When asked by a reporter if she was saying Chinese people should Americanise their names, she responded: "No, and I didn't choose my words very well evidently from what the transcript, from how the transcript reads."
The Texas Democratic Party has called on Mrs Brown to apologise.
In a letter to Mrs Brown sent Thursday, New York City Councilman John C. Liu called on her to either apologize or resign.
"It's outrageous and insulting for you to suggest it would 'behoove' us to adopt another name, to give up our birthright and a part of our own identity, in order to exercise our right to vote," he said.
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