A meeting of Afghan elders is expected to back a security pact letting thousands of US troops remain after combat operations end in 2014.
Sources said more than 75% favour a deal being signed this year, contrary to what President Hamid Karzai wants.
He told the opening session of the assembly or Loya Jirga that it would be signed after elections due next year.
The US has said it is neither "practical nor possible" to delay the signing.
More than 2,000 elders have been meeting behind closed doors for the past few days.
They are expected to set out their final position on Sunday, when President Karzai makes his closing speech.
Sources have told the BBC that more than three-quarters of the committees which have been discussing the document believe that a swift signing is essential.
The BBC's Karen Allen in Kabul says the past few days have seen tense diplomatic telephone exchanges between US Secretary of State John Kerry and President Karzai.
Washington insists the deal - which has taken months to negotiate - must be signed before the end of this year in order to secure plans for how many troops will remain in Afghanistan beyond 2014.
Opening the four-day grand assembly of elders on Thursday, Mr Karzai urged delegates to approve the new security deal.
He said a number of world leaders - including from Russia, China, and India - were backing the accord, and that it would provide the security Afghanistan needed, as well as the foundation for forces from other Nato countries who were assisting Afghan troops.
But he appeared to set a new condition, saying any pact would not be signed until after presidential elections.
That vote will be held in April. Mr Karzai has served two terms so cannot stand again.
State department spokeswoman Jen Psaki said: "We believe that signing sooner rather than later is essential to give Afghans certainty about their future before the upcoming elections, and enable the United States and other partners to plan for US presence after 2014.
"It is neither practical nor possible for us to further delay because of the uncertainty it would create."
The deal under discussion may see 15,000 foreign troops remain after 2014, although the US says it has not yet taken a decision on any presence.
The Loya Jirga can amend or reject clauses in the pact, though its decisions are not binding. The deal will also have to be approved by parliament.
Security is tight for the meeting after a suicide bombing last weekend near the huge tent where it is being held.
The Taliban has branded the meeting a US-designed plot, and has vowed to pursue and punish its delegates as traitors if they approve the deal.