US media reports quoted the father of a crew member on Thursday as saying the ship was heading to a Kenyan port with an armed guard after a US Navy team boarded the ship, but this could not be confirmed.
Officials from the US Federal Bureau of Investigation (FBI) also said on Thursday they had assigned negotiators to help secure the release of the ship's captain.
Earlier on Thursday, the US Navy warship Bainbridge arrived on the scene and was near the 17,000-tonne Maersk Alabama as crew members negotiated with the pirates for the return of the captain.
Half a dozen other ships that had been patrolling the area, but were hours away when the hijacking occurred, were heading to the scene.
The hijacking was the first involving an American crew off the Horn of Africa.
Kevin Speers, a spokesman for Maersk Line, the US subsidiary of Denmark's AP Moller-Maersk which owns the ship, confirmed that the crew had regained control of the Maersk Alabama after the pirates left the ship with the captain.
He added that the crew were unharmed.
Colin Wright, who identified himself as the third mate on the ship, told the Associated Press news agency by satellite phone that the crew were also trying to negotiate the hostage's release.
Among the ship's cargo are 400 containers of food aid, including 232 containers belonging to the UN World Food Programme that were destined for Somalia and Uganda.
The Maersk Alabama hijacking comes just days after armed men took control of a British-owned ship and after three other ships were seized over the weekend.
There had been a relative lull in the number of hijackings in the first three months of 2009, with eight ships overrun by pirates in the Gulf of Aden, a major shipping route between the southern and northern hemispheres.
Somali pirates last year seized dozens of vessels close to the Somali coastline in their attempt to secure ransom payments, taking hundreds of sailors hostage.
The drop in the number of successful pirate attacks in the past three months was largely due to the increased presence of foreign navies in the area.