CHANDLER, Ariz. -- President Obama and Arizona Gov. Jan Brewer had what was described as an intense encounter on the tarmac after the president arrived in the Grand Canyon state on Wednesday.

Brewer greeted the president as he arrived off Air Force One, as governors often do when the president visits a state, and handed him a letter that she described as a “welcome” note to the state, and an invitation to talk about a comeback for her state.

"He immediately took umbrage, if you will with my book, 'Scorpions for Breakfast,' and was somewhat disgruntled, if you will, by the way he was portrayed," Brewer later told Fox News in an interview with Greta Van Susteren. "He's very thin-skinned."

In the tome, Brewer wrote that the president was “patronizing” in a one-on-one meeting about border security.

At the time, in media interviews she described it in a more positive light, and she told Fox News on Wednesday that she thought the two had simply agreed to disagree. 

Obama, however, disagrees with her portrayal of their interactions and the impression that he didn’t treat her cordially.

The White House says the president is willing to meet with her again and looks forward to discussions about how to improve Arizona’s economy.

Reporters who witnessed the exchange Wednesday more closely described that the two spoke “intensely” and chatted longer than the typical meet and greet on the runway. The pool report also says Brewer was pointing her finger at him at one point and they seemed to also being talking over one another in another instance. Obama also appears to be walking away from her while they were still talking, and she later said didn’t finish her sentence in the exchange.

"And I said... 'I have all the respect in the world for the office of the president, but the book is what the book is,'" Brewer told reporters after Wednesday’s encounter.

She said when asked that she wasn’t thrown off by the president bringing up the book, but that she thought they probably should have talked about something like jobs.

Lastly, the governor summed up their relationship, saying, "We both love our country...and we just fundamentally disagree in how getting there. In closing I asked him...I said my offer still stood to come back and go to the border with me and that I would buy him lunch." 

The two have disagreed in the past over how to handle issues related to border security, and also Arizona’s controversial and stringent immigration law.