Thompson also reported attempts to jam satellite feeds of the British Broadcasting Corporation services into Iran and to swamp its London phone lines with automated calls.
In extracts from a speech he will make later on Wednesday, Thompson stopped short of explicitly accusing Tehran of being behind the cyber-attack, but he described the coincidence of the attacks as "self-evidently suspicious".
Last month, Thompson accused Iranian authorities of arresting and threatening the families of BBC journalists to force them to quit the Persian news service.
"It now looks as if those who seek to disrupt or block BBC Persian may be widening their tactics," he said in the extracts of his speech, which the BBC released in advance.
"There was a day recently when there was a simultaneous attempt to jam two different satellite feeds of BBC Persian into Iran, to disrupt the Service's London phone-lines by the use of multiple automatic calls, and a sophisticated cyber-attack on the BBC," he said.
"It is difficult, and may prove impossible, to confirm the source of these attacks, though attempted jamming of BBC services into Iran is nothing new and we regard the coincidence of these different attacks as self-evidently suspicious," he added.
There was no immediate comment from Iranian authorities.
BBC Persian staff provide Farsi-language TV, radio and online services. Few Western journalists are permitted to work in Iran where the hardline Islamic government views much of the foreign media with suspicion.
The BBC's TV service has often been jammed and is only available to owners of illegal satellite receivers.
Thompson said he did not want to give any more details of the latest incidents but added: "We are taking every step we can, as we always do, to ensure that this vital service continues to reach the people who need it."
All BBC Persian service staff work outside Iran, and Thompson has accused Tehran authorities of instead arresting and intimidating their relatives who still live inside the country.
(Writing by David Stamp)