Under the update, USA immigration authorities will view applicants' use of government assistance programs as a negative mark against them when considering their eligibility for a temporary visa or a green card.
Mr Cuccinelli said the words "certainly are" still part of the American ethos when challenged in an NPR interview but then went on to give a revised version of them.
The report says the new rules will take effect from mid-October, 2019, and that they are likely to "have a wide impact in states with large immigrant communities, like California, where many families are made up of citizens and noncitizens alike".
"Give me your exhausted and your poor who can stand on their own two feet and who will not become a public charge", Ken Cuccinelli, the acting director of U.S. Citizenship and Immigration Services, said Tuesday, twisting Emma Lazarus's famous words on a bronze plaque at the Statue of Liberty.
Lazarus' poem reads: "Give me your exhausted, your poor, your huddled masses yearning to breathe free, The wretched refuse of your teeming shore".
"Give me your exhausted and your poor - who can stand on their own two feet and who will not become a public charge", he said.
USA law has long discouraged immigrants from ending up on the public dole, and the new rules law out a method for doing that. "That plaque was put on the Statue of Liberty at nearly the same time as the first public charge was passed - very interesting timing". "Send these, the homeless, tempest-tost to me, I lift my lamp beside the golden door!"
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"All immigrants who can stand on their own two feet, self-sufficient, pull themselves up by their bootstraps", would be welcome, he added.
After the host asked if the policy "appears to change the definition of the American dream", he said: "We invite people to come here and join us as a privilege".
US President Donald Trump has said that he wants to introduce a merit-based immigration system that would favour those who are well-educated or have special skills.
On Tuesday, Cuccinelli described the public charge as a "burden on the government".
The Trump administration has taken the toughest stance against legal and illegal immigration of any presidency in modern times.
The Democratic led House Homeland Security Committee condemned Mr Cuccinelli's revision in a tweet, calling the words "vile and un-American".
Cuccinelli, a former Virginia attorney general, has long held a hard-line stance against immigration and asylum policies.
Give me your exhausted, your poor ... but not too poor.