Wikileaks files reveal secret US-Yemen bomb deal
The files also reveal that Mr Saleh rejected an offer to deploy US ground forces in Yemen.
The US fears Yemen has become a haven for al-Qaeda in the Arabian Peninsula.
The cables detail how Mr Saleh claimed responsibility for two US air strikes in December 2009, according to the Guardian .
A few days after the second attack on 24 December, Mr Saleh told the then head of US central command, General David Petraeus: "We'll continue saying the bombs are ours, not yours."
On 21 December, US ambassador Stephen Seche reported in a dispatch that "Yemen insisted it must 'maintain the status quo' regarding the official denial of US involvement."
Mr Seche quotes Mr Saleh as saying that he wanted operations to continue "non-stop until we eradicate this disease".
The messages are among more than 250,000 US cables obtained by the whistle-blowing website Wikileaks.
The files are released in stages by Wikileaks, and details are also being published in the Guardian, the New York Times and other papers around the world that investigated the material.
According to the files released on Friday, Gen Petraeus had flown in to Yemen's capital Sanaa to tell Mr Saleh that the US would also allow its ground forces to be deployed in Yemen on counter-terrorism operations.
Mr Saleh rejected the offer, although he had told President Barack Obama's national security adviser, John Brennan, in September 2009 that he would give the US full access.
"I have given you an open door on terrorism," Mr Saleh is quoted in a US cable after the meeting with Mr Brennan.
Al-Qaeda in the Arabian Peninsula is suspected of having launched a number of attacks on targets in the West, including failed plots to bomb several cargo airliners in October.
The cables also reveal Mr Saleh to be an erratic partner in negotiations, the Guardian reports.
US security officials who met Yemen's long-standing leader in the course of 2009 described him as "petulant" and "bizarre".
After one meeting with Mr Brennan, the US ambassador reported that Mr Saleh had been "in vintage form". Mr Seche wrote that the President was "at times disdainful and dismissive", while he was "conciliatory and congenial" on other occasions.
Mr Saleh told Mr Brennan that should the US not help Yemen, "this country will become worse than Somalia".