The commissioner's warning comes at a time of unprecedented scrutiny of the behaviour of American tech giants such as Google and Facebook.

Mark Zuckerberg, the Facebook founder, was forced last week to apologise for a data breach affecting 50 million users. Cambridge Analytica, the data analytics firm, is alleged to have used the data while working for the Trump campaign. Last week, EU heads of state and governments agreed to a joint statement calling for "social networks" to guarantee "full protection of citizens' privacy and personal data".

The scandal has encouraged those who claim the companies are as dominant as Standard Oil and AT&T were. Both companies were broken up after falling foul of antitrust legislation.

Mrs Vestager, who was recently labelled "the Eurocrat who makes corporate America tremble", has won plaudits in Brussels for her willingness to take the fight to companies such as Google, Apple and Amazon.

"There is no ban on success in Europe," she said. "You get to be dominant and you get a special responsibility that you don't destroy the already weakened competition.

"We have proven their dominance in search and we have found they have misused this dominance to promote themselves and diminish competitors."

As head of the European Commission's antitrust department, Mrs Vestager has the power to levy billion-euro fines on the world's biggest companies.

Since taking the job in 2014, she has ordered Ireland to claw back £11 billion from Apple and Luxembourg £218 million from Amazon in back taxes after finding they were given illegal tax benefits.

Google has presented a plan of action to remedy the anti-competitive behaviour, which antitrust officials are still scrutinising but last month, a group of Google's opponents said its actions were "essentially meaningless" without breaking the internet giant up.

On the other cases, she said that "what we do say, will have to stand up in court", with no room for conspiracy theories.

Mrs Vestager rejected any suggestion that she was targeting tech companies," adding: "I think the motives for illegal behaviour are the same for any kind of company. Money, fear, power - these motives have been the same across centuries."

She said it was also difficult to use old regulation to oversee some of these new companies.

"There is willingness in Europe to say this is amazing," Mrs Vestager said, "but we are also ready to regulate, if necessary."