"No one has heard from Mullah Omar in sometime and, given attacks like last week's when some Taliban threw acid on girls going to school, many don't seem to show a willingness to negotiate," the U.S. official said.
"We're not seeing any indication from Mullah Omar that he is ready to renounce violence, break all ties to al-Qaida and support the Afghan government and constitution."
Karzai has vowed to offer protection to Omar if the militant leader agrees to take part in peace talks, CNN said.
The Afghan president paired his offer to the Taliban official with a stern warning to possible international opposition, to remove him from office or leave Afghanistan.
Notable deaths of 2014 [PHOTOS]