The incoming US president plans to send 8,000 more troops to Afghanistan
Up to 2,000 extra British troops are likely to be sent to Afghanistan next year, the BBC has learned.
Ministers are considering sending reinforcements to Afghanistan to meet an expected request from Barack Obama, when he becomes US president next year.
In talks, Afghan leaders told Gordon Brown more troops were needed as it emerged two Royal Marines were killed on Tuesday in the south of the country.
The Ministry of Defence said it had not received a request for extra troops.
The last two days in Afghanistan have been marked by bloody fighting, and at least 21 military and civilian deaths.
At Downing Street, during a visit to London, Afghanistan's President, Hamid Karzai, told Gordon Brown all efforts were being made "to bring violence down" after the Afghan foreign minister urged Britain to send in more troops.
Mr Karzai told BBC News: "British troops have been in the very difficult part of Afghanistan, in the most difficult part of the country. They have suffered, they have sacrificed lives in Afghanistan.
"The Afghan people are very grateful for what Britain has done in Afghanistan."
He added: "If we need more troops to add to security, to close the borders... [to] the entry of extremists and terrorists, the exit of narcotics, well, yes, bring more troops."
But among Britons, there appears to be a public appetite to pull out.
A BBC-commissioned poll suggested nearly 70% favoured bringing the troops home.
However, the US President-elect Barack Obama has made it clear he is ready to commit at least two more American brigades, the equivalent of about 8,000 extra troops.
And it is expected he will ask Nato allies to strengthen their numbers too.
The UK already has 8,100 troops in Helmand province and British ministers have publicly argued any extras should come from elsewhere in Europe in order to share the burden more fairly.
But the BBC's diplomatic correspondent James Robbins said that privately ministers and officials conceded the new president would still ask for a greater British fighting effort.
He said they also made it clear that no government would want to say no to President Obama early in his term of office, particularly given his huge following.
Our correspondent said some British officials accepted it might be necessary to commit up to 2,000 more soldiers next year.
But he said it could be made easier by the expected withdrawal from Iraq and the net effect could still be a reduction in force levels in the two countries combined.
The Ministry of Defence told the BBC: "We have not received a request for extra troops and the figure of 2,000 is not one we or Number 10 recognise.
"Whereas we keep our force levels under constant review, based on military advice, there is no further intention to increase force levels at this time."
The combined death toll of British service personnel in Iraq and Afghanistan has now reached 300.
In total, 124 of them were in Afghanistan, including the two marines killed on Tuesday when their vehicle was hit by an explosive device in Helmand.
In a rare radio interview earlier, the Taleban's senior spokesman called for all foreign forces to leave Afghanistan.
Speaking by telephone from a secret location in the region, Zabihullah Mujahid derided Barack Obama's plans to deploy more troops, saying they would not defeat the Afghan insurgency.