The UK could "drift towards" exiting the EU if problems are not addressed, David Cameron is set to warn.
The prime minister postponed a long-awaited speech on the UK's relationship with Europe to respond to the hostage crisis in Algeria.
In extracts released in advance, Mr Cameron said he wanted to set out a "positive vision" for the future of the EU in which Britain would play a part.
Meanwhile, President Obama said he wanted a "strong" UK in a "strong" EU.
Mr Cameron had been planning to address an audience of Dutch business leaders, with a speech which would have been closely watched by other European leaders, the business community and supporters and critics within his own party.
But this was postponed after concerns grew about the fate of a number of British hostages being held in a desert gas complex in Algeria. No new date has yet been set for the speech.
Mr Cameron had been expected to set out his vision for the UK's future role in the European Union, including the prospect of a referendum.
The Conservative leader had been under pressure from many of his MPs to give a binding commitment to a vote on Europe when he delivered his speech in Amsterdam.
Extracts from Mr Cameron's speech released on Thursday night reveal he had intended to set out a "positive vision for the future of the European Union. A future in which Britain wants, and should want, to play a committed and active part".
He planned to stress the EU's structures were undergoing "fundamental change", adding: "There is a gap between the EU and its citizens which has grown dramatically in recent years and which represents a lack of democratic accountability and consent that is - yes - felt particularly acutely in Britain."
"If we don't address these challenges, the danger is that Europe will fail and the British people will drift towards the exit," he was to say.
"I do not want that to happen. I want the European Union to be a success and I want a relationship between Britain and the EU that keeps us in it."
According to Mr Cameron, the austerity measures taken to deal with the crisis in the eurozone have given added urgency to the issue of the EU's democratic legitimacy.