The statement, issued through a spokesman on Tuesday, was a riposte to US Major-General Eric Olson, who said that Taliban attacks had "decreased dramatically", and that Umar was no longer able to exercise control over his fighters.
"This is part of America's psychological war aimed at demoralising the Taliban and creating rifts amongst them," Umar said in a statement read by spokesman Abd al-Latif Hakimi in Kabul.
"I have support not only from Muslims in Afghanistan, but from around the world," said Umar, who once declared himself to be the "Commander of the Faithful".
Taliban fighters have kept up an insurgency since being driven from power in late 2001 for giving shelter to al-Qaida and its leader, Usama bin Laden, following the 9/11 attacks on the United States.
Hakimi also claimed the Taliban was responsible for the killing of a British man in Kabul on Monday night.
The Taliban spokesman said Umar had told his deputy, Mullah Ubaid Allah, to initiate an attack that would send a clear signal to the US that the Taliban command could order a strike whenever it chose.
But security sources in the Afghan capital doubted the veracity of the statement, saying it appeared opportunist and the Taliban had often made false claims in the past.
"I have support not only from Muslims in Afghanistan, but from around the world"
Mullah Muhammad Umar,
For his part, General Olson said he expected President Hamid Karzai to announce an offer of amnesty to Taliban rank-and-file soon, and said 30 medium-level Taliban had recently surrendered to US-led forces.
Karzai has said his government is in contact with Taliban members but the amnesty offer will not extend to Mullah Umar or up to 150 of his most loyal followers.
Umar's statement said those who had surrendered were bandits and that no "true Muslim will surrender to the infidels".