Kabul, Afghanistan (CNN) -- America's special envoy to the region talked peace and reconciliation with Hamid Karzai in Kabul Saturday, while the Afghan president made it clear that Afghans should be in the driver's seat.
To that end, Karzai, in an address to the Afghan parliament, said the government and its peace council were making every effort to bring an end to the bloodshed.
"The Afghan nation is the owner of the peace process and any peace talks," Karzai said. "No other country or organization has the right to deprive the Afghan nation to this right. Afghanistan is not a place for foreigners to do their political experiments or a laboratory that every few years they test a new political system."
Karzai recently met with the party of warlord Gulbuddin Hekmatyar, head of the militant Hizb-i-Islami.
"It was a positive meeting and happened in a good atmosphere," said a senior Afghan official with direct knowledge of the talks.
The official, who did not want to be identified because the talks were confidential, said Hizb-i-Islami supports the process of negotiating with the armed opposition and more talks are planned.
The meeting with Hekmatyar served as a signal from Karzai to the United States that peace talks encompass other radical groups besides the Taliban.
Hizb-i-Islami has attacked U.S. forces and Hekmatyar is on America's most-wanted list of terrorists. Another group, the Haqqani network, is another controversial branch of the insurgency accused of receiving sanctuary and support from Pakistan. It's not clear what role it will play in the peace process.
U.S. diplomat Marc Grossman is set to meet Karzai again Sunday in discussions aimed at a achieving a peaceful resolution to Afghanistan's conflict. Karzai is expected to reiterate the the importance of Afghan control over a complex process.
"I am pleased to be in Kabul to consult with the government of Afghanistan," Grossman said in a statement.
"The United States stands ready to assist in any way we can an Afghan-led reconciliation process to find a peaceful end to this conflict. I look forward to calling on President Karzai and discussing next steps."
Karzai's spokesman Aimal Faizi declined to give details of the talks with Grossman.
But another senior Afghan official, who also wished to remain anonymous, said Saturday's conversation went well and included "all aspects" of the peace process.
Grossman has been meeting secretly with Taliban negotiators for more than a year. Last week, U.S. senior administration officials said the United States could inch closer toward peace talks with the Taliban if Karzai blesses the negotiations.
"We don't have any idea standing here today what the outcome of such discussions could be," U.S. Secretary of State Hillary Clinton said last week. "I think all of us are entering into it with a very realistic sense of what is possible. And that includes, of course, President Karzai and his government, which after all bear the ultimate responsibility and the consequences of any such discussions."
Clinton acknowledged last week that discussions about opening up a Taliban office in Qatar and transferring some Taliban prisoners held at Guantanamo Bay were part of the U.S. support for Afghan reconciliation efforts.
She stressed that nothing had been concluded, but said she was sending Grossman to Kabul and Qatar for further consultations. Diplomatic sources said that depending on the outcome of the talks with Karzai, Grossman could have another meeting with the Taliban.
Secret talks between the United States and the militant Taliban began in November 2010, sources said. Senior U.S. officials and diplomatic sources said German officials brokered initial meetings with Tayeb al-Agha, an aide to Taliban leader Mullah Omar. The sources said after a series of meetings, Agha was able to prove his good faith as an interlocutor with key Taliban leaders.
The idea is for the talks to be Afghan-led, senior officials said, with as much U.S. participation as the Afghans need and want.
The talks would be the product of what officials call "confidence-building measures," such as the opening of the office in Qatar's capital Doha and a Taliban renunciation of terrorism in exchange for the release of five Taliban members being detained at Guantanamo Bay.
Clinton said last week no transfers were imminent, but Grossman's talks in Kabul are expected to address the possible release of the detainees.
Once described as preconditions for peace talks to occur, Clinton now says insurgents must lay down their arms, accept the Afghan constitution and end ties with al Qaeda as "necessary outcomes" of the discussions.
Grossman's visit to Afghanistan also comes days after Pakistan's government, embroiled in a squabble between civilian and military leaders, declined a visit from Grossman, according to a senior Pakistani government official.
"His visit could fuel anti-American sentiments and create trouble for the government, which is already surrounded by storms," the official said Wednesday.
The official asked not to be named because he is not authorized to speak to the media on this subject.
A U.S. State Department official said Grossman wanted to go to Pakistan, but the Pakistani government declined for the time being.
CNN's Nick Paton Walsh and Elise Labott and journalist Nasir Habib contributed to this report.