His remarks echoed Donald Trump's racist statement that the Ms Omar and three other congresswomen of colour should "go back" to the "totally broken and crime infested places from which they came."
Mr Paul made the remarks in an interview with Breitbart News at the Turning Point USA conference in Washington last week.
Mr Trump's tweets earlier this month about the four congresswomen of colour, known as "the squad" prompted the House of Representatives to pass a resolution condemning his remarks. Three of the congresswomen were born in the United States, while Ms Omar is a naturalised US citizen who was born in Somalia.
Mr Paul defended the US president's comments, saying of Ms Omar, "I'm sort of dumbfounded how unappreciative she is of our country".
"While I'm not saying we forcibly send her anywhere, I'm willing to contribute to buy her a ticket to go visit Somalia, and I think she could look and maybe learn a little bit about the disaster that is Somalia – that has no capitalism, has no God-given rights guaranteed in a constitution, and has about seven different tribes that have been fighting each other for the last 40 years," Mr Paul said.
"And then, maybe after she's visited Somalia for a while, she might come back and appreciate America more."
A spokesman for Ms Omar declined to comment on Mr Paul's remarks. Mr Paul's office did not immediately respond to a request for comment.
On Sunday, after reports of Mr Paul's comments began to circulate widely, Ms Omar retweeted a post by comedian Tom Arnold appearing to make light of a 2017 incident in which the Kentucky Republican was assaulted by a neighbour.
The retweet sparked outrage among conservatives, including the president's son, Donald Trump Jr.
"Congresswoman Ilhan Omar retweeting calls for political violence against @randpaul," Trump Jr said in a tweet. "I'm not surprised, and look forward to the forthcoming silence from the media on the issue."
Ms Omar, a freshman and one of two Muslim women in the House, have been among the most vocal critics of Mr Trump's policies and language.
In a New York Times op-ed last week, she said the president "has used overtly racist rhetoric to strike fear and division in communities of colour and religious minorities across the country".
She also described her journey with her family to the US in the hope of a better life – and said that the chants of "Send her back!" by Trump supporters at a recent rally in response to the president's criticism of her prompted her to "remember the horrors of civil war in Somalia that my family and I escaped".
"The ideals at the heart of our founding – equal protection under the law, pluralism, religious liberty - are under attack, and it is up to all of us to defend them," she wrote.
In the Breitbart interview, Mr Paul said he has met "a lot of people" in Kentucky who are refugees – "some from Somalia, some from Bosnia" – and "most of them are thankful".
"I've met people who have come here from behind the Iron Curtain, that got away from communism," he said. "They're some of the best Americans we have, because they really appreciate how great our country is."
Ms Omar, by contrast, is "about as ungrateful as you can get", Mr Paul said.
"And then I hear Representative Omar say, 'Oh, America's a terrible place! I thought there'd be justice, and there's no justice here,'" Paul said. "It's like, well, she came here and we fed her, we clothed her, she got welfare, she got school, she got healthcare, and then, lo and behold, she has the honour of actually winning a seat in Congress, and she says we're a terrible country?"
While Ms Omar has been critical of Mr Trump, she has not described the US as a "terrible country".