Brown had blocked the honorary knighthood for Jobs in 2009 after the Apple boss snubbed an appearance at the then ruling Labour Party's annual conference, The Daily Telegrap' quoted a former senior Labour MP as saying.
Jobs, now 55, was put forward for the honour by the MP for services to technology.
Apple is credited with revolutionising the industry in recent years with its iPhone and iPad devices, which have unleashed a boom in net usage.
The former MP, who left Parliament at last election, said: "Apple has been the only major global company to create stunning consumer products because it has always taken design as the key component of everything it has produced.
"No other chief executive officer has consistently shown such a commitment."
Apple was aware of the proposal, he said, adding it had reached the final stages of approval, but was rejected by 10 Downing Street.
The former MP has claimed that he was told by Downing Street that the decision was related to a failed attempt to attract Jobs to Labour's annual conference as his appearance would have been viewed as a coup for Brown.
A spokeswoman for the former Prime Minister, however, said: "Mr Brown did not block a knighthood for Steve Jobs." Jobs is currently on medical leave.
"He has battled, and beaten, pancreatic cancer once before. He also had a liver transplant in 2009."