David Cameron sent an intimate text message to Rebekah Brooks thanking her for a “fast, unpredictable and hard to control but fun” ride on one of her family's horses, it emerged last night.

The message, which could be read as suggestive, and another from Mrs Brooks to Mr Cameron were disclosed to Lord Justice Leveson but were going to be kept secret, sparking claims of a cover-up by the inquiry and Downing Street.

They were revealed by The Mail on Sunday last night just weeks before Lord Justice Leveson is due to publish his report into the phone-hacking scandal which triggered Mrs Brooks' resignation from Rupert Murdoch's News International.

Chris Bryant, the Labour MP, has been pressing the Prime Minister in the Commons to reveal details of what he called "salacious" emails and texts between him and Mrs Brooks, whose husband, Charlie, is an old friend of the PM. Mr Cameron has refused to respond to questions from Mr Bryant, and No 10 has insisted that it has given the Leveson Inquiry all "relevant" documents.

Both texts revealed last night were sent in October 2009, weeks after The Sun switched support from Labour to the Tories. Mr Cameron's text to Mrs Brooks read: "The horse CB [Charlie Brooks] put me on. Fast, unpredictable and hard to control but fun." He signed off "DC".

In the second, sent after his speech to the Tory conference, Mrs Brooks also made an apparently ambiguous remark by writing: "Brilliant speech. I cried twice. Will love 'working together'."

Mrs Brooks is awaiting trial next year on charges of phone hacking and conspiracy to pervert the course of justice.

There are other texts and emails that have been handed over to the Leveson Inquiry but have not been made public because they are not deemed "relevant" to the investigation.

The messages were revealed at the end of another difficult week for the Prime Minister, during which he was defeated by 53 Tory rebels over a cut in the EU budget.

There was also a suggestion that his chief strategist and polling guru, Andrew Cooper, is on the verge of leaving, a move which would cause further upheaval in the beleaguered Downing Street operation following a series of departures of key staff.

Sources said Mr Cooper, the founder of the polling firm Populus, was planning a return to the private sector at a time of deep unhappiness at No 10. Steve Hilton, policy chief James O'Shaughnessy, special adviser Sean Worth, and Tim Chatwin, head of strategic communications, have all left this year, while Andy Coulson was forced to quit nearly two years ago.

It is understood that Mr Cooper had told friends he was planning to leave around the New Year because of a contractual arrangement with Populus. But No 10 last night denied he was leaving, saying "Andrew is going nowhere", leaving open the possibility that he had been persuaded to stay.

In a further blow to the Prime Minister, Liam Fox, the Eurosceptic former defence secretary, is urging Mr Cameron to make a pitch for renegotiation of Britain's membership of the European Union by next October, around the time of the 2013 Conservative Party conference.

Mr Fox wants the Prime Minister to call a referendum well before the 2014 European elections, with a choice of a drastic scaling-down of the UK's role in the EU or total departure – with no option to remain "in" on current terms.

The pressure on Mr Cameron and the No 10 machine intensified on Wednesday when 53 Tory backbenchers helped defeat the Government by voting for a real-terms cut in the EU budget, which will be negotiated on 22 November.

Writing in The Independent on Sunday today, Sir Menzies Campbell, the former Liberal Democrat leader, says Mr Cameron has been "damaged" both at home and in Europe by the vote.

Linking Mr Cameron's situation to that of the dying days of the Margaret Thatcher and John Major governments, Sir Menzies writes: "Defeats like that inflicted last Wednesday damage a prime minister's authority in Britain and his credibility in Europe. Perception is as important in leadership as substance, as I know only too well from my own experience.

"It takes a spurious optimism to claim, as the rebels do, that the screaming headlines of Thursday will somehow fortify the Prime Minister in his budget endeavours. It may not be quite as dire as Browning's perceptive observation 'Never glad confident morning again', but the anti-Europeans in the Tory party haven't just smelled blood: they have tasted it."