Zaslavskiy believes only the FSB, the Russian intelligence agency, had the expertise to administer a toxin slowly enough to harm Dudley but stop far short of killing him.
The alleged poisoning took place two years after the murder of Alexander Litvinenko, who was exposed to radioactive polonium-210 in London, and a decade before the attempted assassination of Sergei Skripal, who was targeted with nerve agent in Salisbury.
Five years after Dudley left Moscow, TNK-BP was sold to the state-owned Rosneft. In return, BP, by then being run by Dudley, was paid a £12 billion lump sum and given a 12.5 per cent stake in Rosneft.
Dudley shook hands on the deal at a ceremony in Russia, attended by Vladimir Putin, the Russian President.
Zaslavskiy said: "The whole idea was to oust Bob Dudley and about 150 Western managers."
Zaslavskiy, who is now the head of research at the Free Russia Foundation, an anti-Putin think-tank in Washington, and a fellow with the Chatham House policy institute in London, went on: "The harassment of Western staff took place on many levels. They followed people; they broke into their apartments. One was warned it wasn't safe to return to Russia.
"With Dudley and a few senior managers, they also started to open different court cases against them personally rather than just BP. That was the first hint to Dudley he should get out. But he wouldn't. He was stubborn. He thought he could run the company because he had the law on his side, so he stuck it out in Moscow.
"Finally they decided to send him a better message by poisoning his food. Obviously we could never prove who exactly poisoned him."
Zaslavskiy added: "I was aware of these claims but I had it confirmed in 2016 by a former colleague who was a close adviser to Dudley. He told me that the poisoning took place in the summer of 2008. They think the poison was put in Dudley's food, maybe by somebody inside the office.
"They did extensive lab tests in Britain and it was confirmed that the poisoning had taken place in Russia. His blood showed signs of poisoning.
"I have now had this confirmed by a good source close to Dudley and I have no reason to doubt it. The reason BP wanted to do the tests was Dudley had felt ill. I don't know the symptoms but I do know they were sufficient for him to get very concerned. It wasn't like regular food poisoning. It wasn't just once. It was happening over weeks.
"Dudley left Russia secretly and quickly after the diagnosis. He left Moscow by the back door. He left early in the morning and he left abruptly. He didn't go back to the London office of BP. He went to a secret location in Europe."
Zaslavskiy suggested that the British Government should have stood up to the Kremlin at the time of the alleged poisoning. Its failure to do so effectively encouraged Moscow to attack Skripal in Salisbury without fear of previous reprisals.
Zaslavskiy added: "I blame some of this on the British Government and MI6 for not standing up to the Russians in the first place. It is astonishing what British security services allowed the Russians to get away with back then.
"The poisoning had to be done by professionals to make sure that Dudley did not die. They just wanted to give him a message. But who definitely carried out the poisoning, I don't know. I don't know the substance. My source wasn't open to talk about this."
BP has declined to comment about the allegation and refuses to say whether Dudley was or was not subjected to a poison attack.
Last week, at a meeting with journalists, Dudley was asked what advice he would give to young executives starting out in the oil business. His response was: "Stay away from Russia."
He added: "I know there are some friends here from Russia - so that's just a joke. I do enjoy working there. I lived there for nine years with my family."
A senior BP source said: "TNK-BP was an extremely successful business financially. It was also extremely noisy at that time. Certainly at various times life was made difficult and it was difficult to operate. But we are not going to comment whatsoever about Bob.
"Certainly pressure was put on him using regulatory means and things like that. But nobody will ever comment about these allegations of poisoning."
