Neither the US nor Nato have confirmed the cause, but witnesses, officials and the Taliban say it was shot down.
The incident is believed to be the biggest single loss of life for US forces in Afghanistan since operations began in 2001.
The Chinook helicopter went down overnight in Wardak province, the statement from President Karzai's office said.
It was returning from an operation against the Taliban in which eight insurgents are believed to have been killed.
A senior official of President Barack Obama's administration said the helicopter was apparently shot down, Associated Press news agency said.
An official with the Nato-led coalition in Afghanistan told the New York Times the helicopter was shot down with a rocket-propelled grenade.
"The president of the Islamic Republic of Afghanistan expresses his sympathy and deep condolences to US President Barack Obama and the family of the victims," the statement from Hamid Karzai said.
Nato's worst Afghan moments
- 6 April 2005 - Chinook crash in Ghazni province kills 15 US soldiers and three civilian contractors
- 28 June 2005 - 16 US troops killed when Taliban bring down Chinook in Kunar province
- 16 August 2005 - 17 Spanish soldiers die when Cougar helicopter crashes near Herat
- 5 May 2006 - 10 US soldiers die after Chinook crashes east of Kabul
- 2 Sept 2006 - 14 UK personnel killed when RAF Nimrod explodes following mid-air refuelling
- 18 August 2008 - 10 French soldiers killed in Taliban ambush east of Kabul
- 6 August 2011 - 31 US special forces and seven Afghan soldiers killed in Chinook crash
The Nato-led International Security Assistance Force has confirmed the helicopter crash but has not released details of casualties or the cause.
ABC News said 25 Navy Seals were on board, plus seven Afghan special forces soldiers, five crew and one interpreter.
AP says most of those killed were from Seal Team Six - the same unit which killed Osama Bin Laden in Pakistan May.
The size of Team Six is not known.
Nato said it was mounting an operation to recover the helicopter and find out why it crashed. It said there had been "enemy activity in the area" where it went down.
A Taliban spokesman said insurgents had brought down the helicopter with a rocket after US and Afghan troops attacked a house in the Sayd Abad district of Wardak where insurgents were meeting late on Friday, Associated Press said.
Sayd Abad, near the province of Kabul, is known to have a strong Taliban presence.
A Wardak government spokesman quoted by AFP news agency agreed with this, saying the helicopter was hit as it was taking off.
A local resident told the BBC Pashto service a rocket hit the helicopter.
"What we saw was that when we having our pre-dawn (Ramadan) meal, Americans landed some soldiers for an early raid," said Mohammad Wali Wardag.
"This other helicopter also came for the raid. We were outside our rooms on a veranda and saw this helicopter flying very low, it was hit by a rocket and it was on fire. It started coming down and crashed just away from our home close to the river."
There are currently about 140,000 foreign troops - about 100,000 of them American - in Afghanistan, fighting the Taliban insurgency and training local troops to take over security.
All foreign combat forces are due to leave Afghanistan by the end of 2014 and some troop withdrawals have already taken place.
Nato has begun the process of handing over control of security in some areas to local forces, with Bamiyan becoming the first province to pass to Afghan control in mid-July.
An increase in US troop numbers last year has had some success combating the Taliban in the south of Afghanistan, but attacks in the north, which was previously relatively quiet, have picked up in recent months.