Grossman's visit comes in the wake of an announcement by Taliban insurgents that they plan to open a political office in Qatar ahead of talks with Washington on ending Afghanistan's 10-year war.
A US official said that if Karzai was agreeable, the talks could open within weeks.
"Efforts have been made to establish an office for the Taliban in Qatar for negotiations," Grossman told a news conference with Afghan deputy foreign minister Jawed Ludin.
Ludin said the Afghan government supported the plan for a Qatar office, adding that a Qatar government delegation would soon visit Kabul.
Grossman said that "for negotiations with the Taliban, we also need a clear statement from the Afghan Taliban -- they should renounce violence and cut their relations with Al-Qaeda."
He reiterated that no decision had been made by Washington on the Taliban's demand for the release of members of the hardline Islamist group held in the US military prison at Guantanamo Bay in Cuba.
Secretary of State Hillary Clinton sent Grossman to Kabul to discuss the Qatar development with Karzai, who was reportedly concerned that he would be sidelined by talks in the Gulf state.
Washington has consistently said that any talks with the Taliban to end the war could only take place with the agreement of the Afghan government, which eventually should lead the process.
"As Secretary Hillary Clinton said last year we are in close cooperation with the government of Afghanistan on peace talks," Grossman said.
The US envoy, who had planned to visit neighbouring Pakistan ahead of his trip to Kabul but was snubbed by Islamabad amid a drastic deterioration in ties, said there could not be "a comprehensive peace process unless Pakistan participate".
US officials have deep concerns about the role of Pakistan, with many believing its intelligence services maintain ties with the Taliban inside Afghanistan.
Grossman said he would visit Qatar on Monday.
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@j. von hettlingen: Pakistan is always going to keep fiddling , as they simply have no intention of leaving Afghans to make their own decisions. They have developed a mentality that its a another Province of Pakistan or at least they should decide as to who Governs the Country, regardless of their dire state of affairs !!! Their Foreign Policy decision makers have actually lost the plot, to the extent they will find it difficult for their very basic needs in near future, let alone survival as in its present form.
Maybe Marc Grossman might not be as tough as his predecessor, the late Richard Holbrooke. Given the same circustances, I wonder if the latter would have been snubbed, were he still be in office. No doubt the U.S. has deep concerns about Pakistan's influence on the Taliban in Afghanistan. The increasing attacks on the ISAF show once more the instability in the country. What happens after 2014? There's a saying: when the cat's away the mice will play.