The report said that Saudi Arabia's quest has often been set in the context of countering Iran's atomic programme and it is now possible that the country might be able to deploy such devices more quickly than the Islamic republic.
The report, citing a NATO official, said that nuclear weapons made in Pakistan on behalf of Saudi Arabia are now sitting ready for delivery.
It quoted Amos Yadlin, a former head of Israeli military intelligence, as saying last month at a conference in Sweden that if Iran got the bomb, "the Saudis will not wait one month”.
“They already paid for the bomb, they will go to Pakistan and bring what they need to bring," BBC Newsnight quoted him as saying.
In 2009, King Abdullah of Saudi Arabia warned visiting US special envoy to the Middle East Dennis Ross that if Iran crossed the threshold, "we will get nuclear weapons".
After that Saudi Arabia has sent the Americans numerous signals of its intentions, the BBC programme said.
"I do think that the Saudis believe that they have some understanding with Pakistan that, in extremis (sic), they would have claim to acquire nuclear weapons from Pakistan," Gary Samore, until March 2013 President Barack Obama's counter-proliferation adviser, was quoted as saying on the Newsnight.
The story of Saudi Arabia's acquisition of missiles capable of delivering nuclear warheads over long ranges goes back decades, the report said.
The Saudis secretly bought dozens of CSS-2 ballistic missiles from China in the late 1980s.
They deployed these sophisticated rockets some 20 years ago.
Defence publisher Jane’s revealed the existence of Saudi Arabia’s third intermediate-range ballistic missile site, approximately 200 km southwest of the country's capital Riyadh.
Allegations of a Saudi-Pakistani nuclear deal started to circulate even in the 1990s, but were denied by Saudi officials.
Western intelligence agencies have accused Pakistani scientist Abdul Qadeer Khan of selling atomic knowhow and uranium enrichment centrifuges to Libya and North Korea.
According to the BBC report, the Saudi government is concerned about Iran's nuclear-proliferation programmes and resented the removal of Saddam Hussein and had long been unhappy about US policy on Israel.
The Pakistan foreign ministry has described the story as "speculative, mischievous and baseless", BBC said.
"Pakistan is a responsible nuclear weapon state with robust command and control structures and comprehensive export controls," the Pakistan foreign ministry said.
The Saudi embassy in London, in a statement, has pointed out the UN's "failure to make the Middle East a nuclear free zone is one of the reasons the Kingdom of Saudi Arabia rejected the offer of a seat on the UN Security Council".