The man, identified as 52-year-old Californian construction worker Gary Brooks Faulkner, said he wanted to cross over into the nearby Afghan province of Nuristan because he had "heard bin Laden was living there", according to officer Mumtaz Ahmad Khan.
He was picked up in a forest in the Chitral region late on Sunday, he said.
"We initially laughed when he told us that he wanted to kill Osama bin Laden," said Khan. But he said when officers seized the pistol, the sword, a dagger and night-vision equipment, "our suspicion grew."
He was questioned Tuesday by intelligence officials in Peshawar, the main northwestern city.
Faulkner told police he visited Pakistan seven times, and this was his third trip to Chitral, which is a mountainous region that attracts adventurous Western tourists and hikers. Unlike much of northwestern Pakistan, it is considered relatively safe for foreigners.
Police alleged the American intended to travel to the eastern Afghan region of Nuristan, just across the border.
Chitral and Nuristan are among several rumored hiding places for the al-Qaida leader, who has evaded a massive U.S. effort to capture him since 2001. The focus of that hunt has been along the Afghan-Pakistan border.
Khan said Faulkner was also carrying a book containing Christian verses and teachings.
When asked why he thought he had a chance of tracing bin Laden, Faulkner replied, "God is with me, and I am confident I will be successful in killing him," said Khan.
Faulkner arrived in the Chitrali town of Bumburate on June 3 and stayed in a hotel there.
He was assigned a police guard, as is quite common for foreigners visiting remote parts of Pakistan. When he checked out without informing police, officers began hunting for him, said Khan.
U.S. Embassy spokesman Richard Snelsire said the mission had received notification from Pakistani officials that an American citizen had been arrested. He said embassy officials were trying to meet the man and confirm his identity.