The group discussed the role with several people but chose Paul because he was viewed as best complementing Rubio's response, a spokesman for the tea party umbrella group said, declining to confirm whether Lee was first up for consideration.
Lee said he "just declined" the offer when asked.
"I think at the end of the day the response is more about identifying what our priorities are rather than the personalities, the people who are delivering the response," Lee said.
Lee wasn't the only one who panned the president's speech.
Rep. Chris Stewart, Utah's newest member of Congress attending his first State of the Union, said he was disappointed with it, arguing that Obama glossed over the country's massive debt and out-of-control spending.
"The grave challenge of our day is our debt and our spending; it's controlling the size of the government and it's reinvigorating the economy and I just don't think the president was willing to address it in a meaningful way," Stewart said.
Rep. Rob Bishop, who watched from home, praised Obama for mentioning the significance of families but there were other parts Bishop didn't like.
"Some things actually scared the crap out of me," Bishop, a former school teacher, said in noting the president's call for preschool for all. "The education part especially brings chills to my bones when I think what that could be."
Rep. Jim Matheson, D-Utah, said it's always good for the president to lay out his vision for the country, even if it doesn't mean Congress will follow his lead. Actions are more important than words, the congressman said.
"I'm not looking for a sea change tomorrow morning in terms of what's going on around here," Matheson said.
Emily Andrews contributed to this story.