WASHINGTON  President Obama and his advisers have decided to significantly expand Afghanistan’s security forces in the hope that a much larger professional army and national police force could fill a void left by the central government and do more to promote stability in the country, according to senior administration and Pentagon officials.

A plan awaiting final approval by the president would set a goal of about 400,000 troops and national police officers, more than twice the forces’ current size, and more than three times the size that American officials believed would be adequate for Afghanistan in 2002, when the Taliban and Al Qaeda appeared to have been routed.

The officials said Mr. Obama was expected to approve a version of the plan in coming days as part of a broader Afghanistan-Pakistan strategy. But even members of Mr. Obama’s national security team appeared taken aback by the cost projections of the program, which range from $10 billion to $20 billion over the next six or seven years.

By comparison, the annual budget for the entire Afghan government, which is largely provided by the United States and other international donors, is about $1.1 billion, which means the annual price of the program would be about twice the cost of operating the government of President Hamid Karzai.