The male suspect had not made any threats, KXAS said, and officials do not believe he intended to harm the former first couple.
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Officials told KXAS that the suspect went to a friend's house in Bush's gated neighborhood to show off his Plymouth Barracuda.
The man told investigators he ran onto the Bushes' yard when his gas pedal became stuck.
No one was injured, KXAS reported.
“President and Mrs. Bush are OK," Bush spokesman David Sherzer said in a prepared statement. "They appreciate the work of the United States Secret Service and Dallas Police Department in responding to the incident.”
The Bushes live on Daria Place, a public street but gated and protected by the Secret Service detail assigned to the former president.