Officials say Scott Stockert - who is from the state of North Dakota - was planning to snatch one of the dogs, either Bo or Sunny.
Police in Washington DC found weapons in Mr Stockert's vehicle, including a shotgun, a rifle and a machete.
Agents say he made several outlandish claims during his arrest, including that he intended to run for president.
He also said he was the son of former President John F Kennedy and actress Marilyn Monroe, according to court documents.
Mr Stockert has been charged with violating the District of Columbia's gun laws.
The First Family acquired Bo in 2009, with Sunny, referred to by the family as Bo's "little sister", following in 2013.
US police have arrested a man who allegedly plotted to kidnap one of President Obama's two pet dogs.