By Yeganeh Torbati

WASHINGTON (Reuters) - The United States suggested on Tuesday it could cut U.S. aid to Pakistan or downgrade Islamabad's status as a major non-NATO ally to pressure the South Asian nation to do more to help it with the war in Afghanistan.

A day after President Donald Trump committed to an open-ended conflict in Afghanistan and singled out Pakistan for harboring Afghan Taliban insurgents and other militants, U.S. Secretary of State Rex Tillerson said Washington's relationship with Pakistan would depend on its help against terrorism.

"We are going to be conditioning our support for Pakistan and our relationship with them on them delivering results in this area," Tillerson told reporters.

U.S. officials are frustrated by what they see as Pakistan's reluctance to act against groups such as the Afghan Taliban and the Haqqani network that they believe exploit safe haven on Pakistani soil to launch attacks on neighboring Afghanistan.

Pakistan denies it harbors militants fighting U.S. and Afghan government forces in Afghanistan.

Tillerson said the United States could consider withdrawing Pakistan's status as a major non-NATO ally, which provides limited benefits such as giving Pakistan faster access to surplus U.S. military hardware, if cooperation did not improve.

"We have some leverage that's been discussed in terms of the amount of aid and military assistance we give them, their status as non-NATO alliance partner - all of that can be put on the table," he said.

In a televised speech on Monday offering few specifics, Trump promised a stepped-up military campaign against Taliban insurgents who have gained ground against the U.S.-backed Afghan government and he singled out Pakistan for harboring militants.

U.S.-backed Afghan forces overthrew the Taliban's hard-line Islamist government in late 2001 because it sheltered al Qaeda chief Osama bin Laden, architect of the Sept. 11 attacks on New York and Washington that year.

U.S. forces have been bogged down since in a war that has vexed three American presidents. About 2,400 U.S. troops have died there in America's longest military conflict.

(For a graphic on U.S. troops and contractors in Afghanistan click http://tmsnrt.rs/2xm6CxQ)

The Afghan government welcomed Trump's speech, but the Taliban said it would make the country a "graveyard for the American empire."

Successive U.S. administrations have struggled with how to deal with nuclear-armed Pakistan, which has a porous border with Afghanistan. Washington fumes about inaction against the Taliban, but Pakistan has cooperated on other U.S. counterterrorism efforts, including against al Qaeda and Islamic State militants.

The Pakistani Foreign Ministry said it was "disappointing that the US policy statement ignores the enormous sacrifices rendered by the Pakistani nation" in fighting terrorism.

"As a matter of policy, Pakistan does not allow use of its territory against any country," it said.

A senior U.S. official said on Tuesday significant measures were under consideration, including possibly sanctioning Pakistani officials with ties to extremist organizations.

INDIA FACTOR

Trump also called for Pakistan's great rival India to play a bigger role in Afghanistan, a prospect that will ring alarm bells for Pakistan's generals.

"Trump's policy of engaging India and threatening action may actually constrain Pakistan and lead to the opposite of what he wants," said Zahid Hussain, a Pakistani security analyst.

The United States has little choice but to use Pakistani roads and air corridors to resupply its troops in landlocked Afghanistan, giving Islamabad considerable leverage. U.S. officials fret that if Pakistan becomes an active foe, it could further destabilize Afghanistan and endanger U.S. soldiers.

Defense Secretary Jim Mattis said on Tuesday he was awaiting a plan from the U.S. military's chairman of the Joint Chiefs of Staff, General Joseph Dunford, before deciding how many more troops to send to Afghanistan.

"When he brings that to me, I will determine how many more we need to send in," Mattis told reporters in Baghdad. "It may or may not be the number that is bandied about."

U.S. officials have said Trump has given Mattis authority to send about 4,000 additional troops to add to the roughly 8,400 already in Afghanistan.

The U.S. Air Force may intensify its strikes in Afghanistan and expand training of the Afghan air force following Trump's decision, its top general told Reuters on Tuesday.

Most U.S. troops in Afghanistan work with a NATO-led training and advising mission, with the rest part of a counterterrorism force that mostly targets pockets of al Qaeda and Islamic State fighters.

(Additional reporting by Idrees Ali in Baghdad, Mirwais Harooni in Kabul, Drazen Jorgic and Syed Raza Hassan in Islamabad, Doina Chiacu in Washington and Phil Stewart aboard a U.S. military aircraft; Writing by Alistair Bell and Arshad Mohammed; Editing by Frances Kerry and Peter Cooney)