The move would also entail a "modest reduction" in troops from Iraq, the newspaper said.
The report comes after George Bush, the US president, announced at a Nato summit last month that the US would significantly increase troop levels in Afghanistan next year.
Gates has also pushed other European member states to provide combat troops and equipment to fill shortfalls in the south of Afghanistan.
The New York Times said the Pentagon now appears resigned to the fact that Nato is unable or unwilling to contribute more troops despite its public pledges.
'Re-Americanisation'
The increasing proportion of US troops in Afghanistan, from about half to about two-thirds of all foreign troops in the country, could result in what one senior administration official described as "the re-Americanisation of the war", the newspaper report said.
"There are simply going to be more American forces than we've ever had there," the report quoted an official as saying.
The US currently has about 34,000 troops in Afghanistan, 16,000 of them under the Nato-led International Security Assistance Force (Isaf) in eastern Afghanistan.
The other 18,000 are involved in counter-terrorism operations and training of Afghan security forces.
A contingent of about 2,500 US marines was deployed to Afghanistan last month to reinforce Nato forces in the south for seven months.