Mr Obama vows US forces will not enter Afghan homes except for “extraordinary circumstances” – a key point of debate.
Mr Karzai urged the 2,000 elders to back the deal, which could see 15,000 foreign troops remain after 2014.
But he said it would not be signed until after elections next year.
The will be held in less than six months’ time, and Mr Karzai has served two terms so cannot stand again.
His office could not confirm to the BBC whether Mr Karzai – or his successor – would sign the pact.
The BBC’s Karen Allen in Kabul says the issue of US raids on Afghan homes has been a key stumbling block in a deal that has taken months to hammer out.
But a draft of the deal was released by Kabul shortly before the grand assembly of elders – or Loya Jirga – started on Thursday.
Our correspondent says that in a dramatic moment as he delivered his speech to the meeting, Mr Karzai produced the letter from Mr Obama which gives an assurance on US raids.
The letter reads: “US forces shall not enter Afghan homes for the purposes of military operations, except under extraordinary circumstances involving urgent risk to life and limb of US nationals.
“We will continue to make every effort to respect the sanctity and dignity of Afghans in their homes and in their daily lives, just as we do for our own citizens.”
It continues: “The US commitment to Afghanistan’s independence, territorial integrity, and national unity, as enshrined in our Strategic Partnership Agreement, is enduring, as is our respect for Afghan sovereignty.”
The Loya Jirga can amend or reject clauses in the agreement, though its decisions are not binding. The deal will also have to be approved by parliament.
Mr Karzai’s statement on the timing of the signature of the document appears to be a new condition, our correspondent says.
The US had wanted the deal to be agreed quickly.
Indeed Mr Obama’s letter says: “We look forward to concluding this agreement promptly.”
The US would have to take into account any amendments that are put forward, and would still have the option of pulling out altogether.
Another key sticking point that Mr Karzai appears to have conceded concerns the jurisdiction for the prosecution of US troops.
The US insistence on immunity from Afghan prosecution for troops has been central to Washington’s demands.
The failure to resolve a similar legal issue in Iraq led to a total withdrawal of US forces.
The US-Afghan draft says: “Afghanistan authorises the United States to hold trial in such cases, or take other disciplinary action, as appropriate, in the territory of Afghanistan.”
According to the draft, the deal will remain in force “until the end of 2024 and beyond”.
Currently the multinational Nato force is due to pull out of Afghanistan from 2014.
Opening the four-day Loya Jirga, President Karzai said the only issue on the table was whether the security agreement would be signed.
A woman delegate shouted from the floor that US troops had spilt too much Afghan blood and should be stopped.
Mr Karzai acknowledged there were difficult issues involved but advised delegates to accept the agreement.
He said that a number of world leaders – including from Russia, China, and India – were backing the deal, 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 Mr Karzai also admitted there was a lack of trust between him and the Americans.
He said: “I don’t trust them and they don’t trust me, the last 10 years has shown this to me. I have had fights with them and they have had propaganda against me.”
The Loya Jirga delegates will now meet in smaller closed-door groups to look at the deal in detail.
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.