The Taliban reacted to Mr Trump's announcement by saying they never wanted to talk to the United States anyway, but one senior member of the group said he suspected efforts would still be made to get negotiations going.
Talking to reporters at the White House on Monday, Mr Trump condemned the militant group for recent carnage in Kabul and said the United States was not prepared to talk now. He pledged to "finish what we have to finish".
His comments suggested he sees a military victory over the Taliban, an outcome that U.S. military and diplomatic officials say cannot be achieved with the resources and manpower he has authorized.
A spokesman for Afghan President Ashraf Ghani said while the government had encouraged the Taliban to talk, the attacks in Kabul, including a suicide bomb attack on Saturday that killed more than 100 people, was a "red line".
"The Taliban have crossed a red line and lost the chance for peace," said the spokesman, Shah Hussain Murtazawi.
"We have to look for peace on the battlefield. They have to be marginalized."
He declined to comment directly on Mr Trump's announcement.
A spokesman for the Taliban, who are fighting to oust foreign forces, defeat the U.S.-backed government and impose their version of Islamic rule, said they never wanted to hold peace talks with the United States anyway.
"Their main strategy is to continue war and occupation," the spokesman, Zabihullah Mujahid, said in a statement.
He said Taliban fighters would respond in kind if the Americans wanted to focus on war: "If you emphasise war then our mujahideen will not welcome you with flowers."
Mr Trump last year ordered an increase in U.S. troops, air strikes and other assistance to Afghan forces.
The U.S. ambassador to the United Nations, Nikki Haley, said this month the strategy was working and pushing the insurgents closer to talks.
That attack followed a brazen Taliban assault on the city's Intercontinental Hotel on Jan. 20, in which more than 20 people, including four Americans, were killed.
The Taliban said the attacks were a message to Mr Trump that his policy of aggression would not work.
Another Taliban member said the United States had been approaching states that have relations with the Taliban to try to get them to push the insurgents to the negotiating table.
"President Trump is saying this for public consumption," the Taliban member, who declined to be identified, said of Mr Trump's rejection of talks. "He and his team are making every effort to bring us to the negotiating table.
"Actually, the latest attack in Kabul awakened President Trump and his puppets in Afghanistan about the capability of the Taliban and their ability to mount big attacks anywhere."
The Taliban refer to the Afghan government as U.S. "puppets".
The United States believes the Haqqani network, a faction within the Taliban, was behind Saturday's bomb blast in Kabul.
It and Afghanistan have long accused Pakistan of supporting the Taliban, and the Haqqani network in particular, as assets to be used in its bid to limit the influence of old rival India in Afghanistan.
This month, Mr Trump ordered big cuts in security aid to Pakistan over its failure to crack down on militants.
Pakistan denies accusations it fosters the Afghan war, and condemned the recent attacks in Afghanistan.
Pakistani officials were not immediately available for comment on Mr Trump's rejection of talks with the Taliban.