The Arizona Republic obtained the letter from Brewer's office, which initially resisted, saying the letter wasn't subject to open-records laws and no copies existed.
"Welcome to Arizona!" begins the letter, which the Republican governor wrote by hand in cursive on official stationery.
"You've arrived in a state at the forefront of America's recovery -- and her future," she writes. "We were at the bottom of the list in job creation. Today, we have a balanced budget and we're in the top 10 for job creation."
She continues, "We both love this great country, but we fundamentally disagree on how to best make America grow and prosper once again. I'd love an opportunity to share with you how we've been able to turn Arizona around with hard choices that turned out to be the right ones. And, of course, my offer to visit the border -- and buy lunch -- still stands."
According to a White House pool report, Brewer "handed him a handwritten letter in an envelope and they spoke intensely for a few minutes. At one point, she pointed her finger at him."
Brewer said that her book, Scorpions For Breakfast, "disturbed" Obama.
President Obama, however, downplayed the incident in an interview with ABC News Thursday. "I think it's always good publicity for a Republican if they're in an argument with me. ...But this was really not a big deal," he said. "Last time we met, she wasn't fully satisfied. But, you know, I think this is a classic example of things getting blown out of proportion."