Hagel travelled to the southern province of Helmand to meet troops a day after further tensions over the security pact that would allow NATO forces to stay in the country after next year.
"I believe there is a role for our coalition partners and the United States, but that depends on the Afghan people," Hagel told US soldiers in a question-and-answer session.
"If the people of Afghanistan want to continue that relationship, then we will."
Meetings with Karzai have been customary over the years for Pentagon chiefs, but Hagel said Saturday after his arrival that he had no plans to meet the Afghan president during his weekend visit.
As President Barack Obama's top national security adviser, Susan Rice, and top diplomat, John Kerry, had already had frank discussions with Karzai urging him to sign the security agreement, Hagel said there was no point in him repeating the US position.
"There's not much I can add in a meeting with President Karzai to what's already been said," he said on Saturday.
Karzai, who visited Iran on Sunday, initially endorsed the Bilateral Security Agreement (BSA), but has since refused to sign and issued fresh demands.
The agreement sets the legal conditions to permit US and other forces to operate in the country beyond 2014.
But without a signed deal, countries ready to send troops to a post-2014 training mission cannot make budget plans or secure political approval, Hagel said.
Karzai has said the signature could take place after elections in April, but Hagel said that would push the timeline into mid-2014 as the polls are expected to result in a run-off vote.
He said a meeting of NATO defence ministers in February would be crucial for military planners and governments "and some answers are going to be required at that NATO ministerial".
There are currently 46,000 American troops and 27,000 soldiers from other coalition countries in Afghanistan, and almost the entire NATO-led force is scheduled to pull out by the end of next year.
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