Litvinenko's widow Marina is interviewed for the programme.
The Kremlin has angrily branded Alexander Litvinenko's widow Marina a "liar" following an interview she has given to Panorama.
She said: "I cannot say exactly that Putin killed my husband but I can say that Putin is behind everything that happens in Russia."
President Vladimir Putin's press spokesman said: "Indirectly she says that Russia has done it. I answer directly that Russia has not done it.
"It is absurd even to think about it."
The programme's investigation into the death of the former Russian security service officer has found that it is likely that there were multiple attempts to poison Litvinenko with Polonium 210 with the first as early as two weeks before he visited the London Sushi Bar on November 1 with academic Mario Scaramella.
The first attempted poisoning may have taken place at the same restaurant in Piccadilly Circus but earlier when Litvinenko met two former KGB men there back on October 16.
The contamination in the restaurant was not found where Litvinenko and Scaramella met on November 1 but at another location, most likely at the seats where Litvinenko met the Russian businessmen Andrei Lugovoi and Dimitri Kovtun in October.
Lugovoi was once a KGB officer and is now a millionaire while his friend Kovtun who is also ex-KGB has lived a shadowy life in Germany.
Scaramella says that he had no knowledge of how the sushi bar became contaminated.
A number of other sites, including a Knightsbridge hotel visited by the two ex-KGB men have already been found to be contaminated.
Panorama has discovered that Aldermaston (the atomic weapons research centre which tried to locate the source of the polonium) got Scaramella's test results wrong and that he wasn't contaminated with polonium.
The first test was based on the wrong type of radiation reading which produced a wildly inaccurate result.
Two subsequent and more substantial tests showed that Scaramella was negative. It was reported that Scaramella had tested positive for a massive dose of polonium.
Scaramella describes the information passed to him from Litvinenko.
He says: "He co-operated for three years with us. He passed such strong information. Some of this information was lethal information shall we say. Other people have been killed for this kind of cooperation."
Asked if he had any part in the poison plot, Scaramella says: "Absolutely not. Not only I have not killed him, of course, but I think my role in this history is very marginal. I simply meet him in the same day something horrible happened."
Litvinenko's widow Marina is interviewed for the programme and says: "I knew about two meetings, with Scaramella, when Sasha (Litvinenko) had lunch, and second meeting, with Lugavoi, very short meeting he told me, very short, just like a few minutes."
Marina explained that Litvinenko met the ex-KGB men on another occasion and that his tea may have been spiked.
She said: "At Millennium Hotel Sasha told me he met Lugovoi and during this meeting he had drunk tea and he said it was tea already served, on the table, and he just took this cup of tea, and he, he didn't finished at all, and how he later said tea wasn't very tasty, because it was cold."
Panorama has also tracked down a potential witness Mikhail Trepashkin, who Scotland Yard have not yet been able to talk to.
Smuggled mobile phone
Trepashkin is a former FSB (Russian Secret Service) officer who was thrown in jail while investigating the Moscow flat bombs which were blamed on Chechen terrorists, although some people such as Litvinenko claimed that the security services were responsible for the attacks.
Speaking on a mobile phone smuggled into his prison by the filmmaker Andrei Nekrasov, Trepashkin explained to him how one of his FSB bosses back in 2001 had Litvinenko in his sights.
"He asked me to go and find out the route to Litvinenko, where he works, his pattern of movements, his meeting places. He said find out what book he's writing. I realised they wanted to send one person to find out his whereabouts and the group will follow."
Panorama also visits Laboratory Number 12, in Moscow once a KGB poison factory. A former Soviet intelligence officer, who preferred to remain anonymous, said about the factory: "It's the laboratory that every year gets it budget to work with radioactive poisons."
Panorama has also obtained a document classified as Top Secret by the Italian government in which Litvinenko accuses Romano Prodi, Italy's prime minister and former president of the European Commission, of being a friend of the KGB.
Before his death Litvinenko collaborated with Scaramella on his KGB mole-hunt in Italy, feeding him with highly sensitive information on KGB operations.
Scaramella tells Panorama that Litvinenko was warned off defecting to Italy.
He said: "His idea when he left Moscow was to arrive in Italy and he mentioned a friend, a colleague in the secret service who told him that you cannot go to Italy because there are some big friends of Russia in this country."
Prodi denies allegations
Asked by Panorama's John Sweeney if he thought Prodi was a KGB spy, Scaramella says: "No I think nothing about that. I simply collected from different people some information about him, about the past, about the present. Some qualified sources, including Litvinenko, told me that some officers in Moscow considered him as their man, KGB man."
Prodi has always denied any allegations that he has links with the KGB.
How To Poison A Spy will be broadcast on BBC One on Monday 22 January at 2030 and will be offered on this site on demand from Tuesday.
My Friend Sasha: A Very Russian Murder will follow on BBC Two at 2320. For three years before his death from poisoning, former KGB agent Alexander Litvinenko had been making a documentary with his friend Andrei Nekrasov about authoritarianism in post-communist Russia.