While that number is expected to hit 68,000 by the end of the year, in line with Obama's order, it was not clear how many troops the commanders seek overall, the Times said.
The report comes as General Stanley McChrystal, the senior US commander, continues his review of the US military's strategy in Afghanistan.
"Afghanistan is very vulnerable in terms of [the] Taliban and extremists taking over again, and I don't think that threat's going to go away"
Admiral Michael Mullen, chairmnan of the US Joint Chiefs of Staff
US forces in eastern Afghanistan are under particular pressure, the US commanders are reported by the Times
to have said.
Major-General Curtis Scaparrotti, who leads United States and Nato forces in that region of the country, is reported by the Times to have told Holbrooke that the Haggani network is gaining ground.
Jalaluddin Haqqani and his son, Sirajuddin, have been linked to al-Qaeda, and are thought to have set up bases in neighbouring Pakistan in order to plan their attacks against US, Nato and Afghan troops.
Admiral Mike Mullen, the chairman of the US Joint Chiefs of Staff, said on Sunday that the situation in Afghanistan is "serious and deteriorating".
"Afghanistan is very vulnerable in terms of [the] Taliban and extremists taking over again, and I don't think that threat's going to go away," he said.
He did not say whether US troop levels should go beyond the number committed to by Obama, but he said that Afghanistan's security should be safeguarded within 12 to 18 months.
"I think it [the security situation] is serious and it is deteriorating, and I've said that over the last couple of years, that the Taliban insurgency has gotten better, more sophisticated," he said.
"We're just getting the pieces in place from the president's new strategy on the ground now … I don't see this as a mission of endless drift. I think we know what to do."