The former Labour Home Secretary claimed Mr Griffiths used “despicable” tactics to try to hang on to a larger office at Westminster in 2001.
He alleged that the Edinburgh South MP threatened to delay a compensation scheme for trawlermen Mr Johnson had been fighting for unless he got his way.
Mr Johnson describes the alleged incident in his new book, The Long and Winding Road.
The two men were both ministers in the Department of Trade and Industry after the 2001 general election and Mr Griffiths, who was the more junior, had been using the office earmarked for Mr Johnson.
Mr Johnson says Mr Griffiths called him for a conversation about the trawlermen’s compensation scheme, which he was responsible for administering and said there was a “problem” in that he was having to move out of the office and “pack all the files away”.
He says Mr Griffiths told him: “After I’ve moved into my new office I’m not sure where this file will be. It may well be stuck at the bottom of the box, which will mean it will be a while before it gets processed”.
Mr Johnson asked if he was saying that the payments would be delayed until he allowed Mr Griffiths to stay in his office. He says Mr Griffiths then “pondered the question, a pained expression on his pinched face, but said nothing”.
“I may have misinterpreted his silence but I don’t think so,” Mr Johnson writes. “As extortion goes it was hardly in the House of Cards category but to me it was equally despicable.”
He adds that, after a few expletives, “suffice to say I got my allocated office and the trawlermen got their compensation in time”.
Mr Johnson told the Cheltenham Literature Festival that disputes about the size of ministerial offices were quite common. He added: “This book is not trying to dig the dirt or have a go at anyone apart from this unctuous little git.”