The Afghan President's initiative includes declaring an indefinite ceasefire between both parties, recognizing the Taliban as a political party, and freeing the prisoners in order to initiate a path towards a peace process that puts an end to more than 16 years of conflict.
Mattis comments came during a trip to Afghanistan that is expected to precede a sharp increase in fighting after U.S. President Donald Trump approved a more aggressive strategy against the insurgents past year that included more U.S. combat advisors and air strikes.
Mattis arrived in capital Kabul on an unaccounted visit - his third since taking charge previous year - during which he met President Mohammad Ashraf Ghani.
"We want the Afghans to lead and provide the substance to the reconciliation effort", he said. "The victory will be a political reconciliation".
He also met NATO-led Resolute Support personnel and other top Afghan government officials to discuss the military effort and the potential for peace between the Taliban and the Afghan government, the U.S. Forces in Afghanistan said in its statement.
He said there were "elements of the Taliban clearly interested in talking with the Afghan government", while admitting that not all the rebels would be prepared to do so. These and other moves boosted the number of US troops in Afghanistan by at least 3,500, to a total of more than 14,000.
Mattis planned to meet with General John Nicholson, the top American commander in Afghanistan, as well as Afghan President Ashraf Ghani. They have so far not responded to Ghani's overture. But Talibans state the sending of soldiers by the U.S. to Afghanistan shows that Washington does not really want the peace.
The Taliban stance is that talks for a conflict-ending compromise must take place with Washington, not Kabul.
For the first time, reporters were asked to withhold publishing their stories until arrival at the US coalition headquarters in Kabul-a security concern dictated by the rocket attack against the Kabul airport a few hours after Mattis visited in September. Neighboring countries are doubtful about America's commitment to a political resolution.
On the military front, US officials assert that years of effort to build a credible and effective Afghan army and air force are beginning to pay off. The number of Afghan combat deaths also is withheld by the Afghan government. Afghan security forces have been stretched thin as fighting escalated after winter ended, the report said.
He said the positive signals had come from small numbers of insurgents.
"We do look toward a victory in Afghanistan - not a military victory", he said.
US intelligence agencies are taking a less optimistic view of the war's likely path in 2018. Afghan forces, while "unsteady", probably will maintain control of most major population centers in 2018, he added.
However, the Taliban appears likely to miss that conference and have so far ruled out direct talks with the Western-backed government in Kabul, which they say is an illegitimate, foreign-imposed regime.