The chairman of the company behind the Daily Telegraph texted David Cameron before the last election to suggest the Conservative leader speak to the editor of his newspaper every day during the campaign to ensure his party's message was getting across in the broadsheet.

Aidan Barclay texted the would-be prime minister in March 2010 after a breakfast meeting with him at the Ritz hotel, which his family also owns, saying he had "spoken to Tony G" – Tony Gallagher, the editor of the Daily Telegraph – and they would "arrange a daily call during the campaign as discussed".

Giving evidence to the Leveson inquiry, in a rare public appearance, Barclay explained he believed that if Cameron "wanted to get the attention of the editor [of the Telegraph] and wanted to get his message across in the most efficient manner, he should make a habit of phoning him on a daily basis and I recommended that's what they should do".

The Telegraph chairman said he did not check "on a daily basis" if the calls took place, and it is understood that the Conservative leader only rang Gallagher three or four times during the election campaign, which eventually saw the Conservatives returned to office in coalition with the Liberal Democrats.

The text was one of a handful sent between Barclay and Cameron, showing that the member of the family that owns the Telegraph used his personal access to try to lobby the Conservative leader, particularly over economic matters and the need for greater financial deregulation. Barclay has known Cameron personally since meeting him during the 2005 leadership campaign.

On 24 May of last year, Barclay texted the now prime minister, suggesting that the "Bank of England announce extension to liquidity scheme" and that banks should have "say five years to implement" the new Basel 3 capital and liquidity rules. The message was signed off "best, Aidan".

Seven minutes later, another message from the Telegraph owner followed: "David, I'm sure you're aware that the credit markets are not good and are likely to get worse, as they all err on the side of caution faced with combination of more regulation Basel 3, more liquidity losses from sovereign debt, the Bank of England support and potential tax all at the wrong time. I hope you don't mind me mentioning it. Regards Aidan."

Sir David and Sir Frederick Barclay, who also own hotels, property, online shopping and logistics companies that employ 20,000 people, bought the Telegraph titles in 2004 for £665m, installing Sir David's eldest son Aidan as the chairman of Telegraph Media Group. The low-profile twins, who divide their time between Brecqhou, off Sark, and Monaco, avoid public appearances, and delegate the running of their portfolio of interests to Aidan and a group of close associates.

Aidan Barclay said the titles were conservative with both a "small and a capital C," and it was clear from the evidence that the Conservative politics of the titles was in keeping with the company chairman, who said he was worried about over-regulation on business and opposed to "high taxation for high earners". But he insisted that he did not "have the habit of interfering with editorial matters".

Instead, he sought influence by asking that the prime minister be emailed an "interesting article" written by Fraser Nelson, the editor of the Spectator – which is also owned by the Barclay family – on the subject of whether increasing the rate of tax on high earners actually increased the amount of money raised by the Treasury overall.

The scion of the family published a list of his meetings with David Cameron and other senior politicians, generally entertaining them at the Ritz hotel, or visiting them at Number 10 or other ministerial offices.

The Ritz breakfast of March 2010 was one of three meetings in the year before the election, including a meeting on 3 November 2009 and a dinner at the Camerons' house on 25 November. Two other meetings followed the election, a meeting and drinks reception on 6 July 2010, and a dinner at Number 10 on 18 November, where Cameron briefly asked the Telegraph chairman for his views on Rupert Murdoch's bid for BSkyB.

Barclay had earlier told the inquiry that he was opposed to the bid because he was worried about the capacity that News Corporation and BSkyB had for cross-selling print and television advertising across the Times, the Sun and Sky television, and the possibility of bundling digital versions of the newspapers to Sky subscribers.

Barclay says he was asked for his view, and outlined his reasons, but the conversation was brief "as it was a dinner where spouses were included". David Barr, inquiry counsel, asked "would it be fair to summarise that as that in very brief and informal circumstances you were able to communicate your views to the prime minister"?

Replying, Barclay reiterated how brief the exchanges were: "Well, first of all, he asked me, and secondly, he didn't comment on what I said. It was kind of a grunt and the conversation moved on."

The Telegraph owner said that he had met Tony Blair five times while he was prime minister, on "relaxed and social" occasions, while there were eight meetings with Gordon Brown when he occupied Number 10.

As with Cameron, Barclay sent Brown articles, and, in 2008 during the financial crisis, sent him a copy of A Monetary History of the United States, an analysis of the Great Depression, written by the monetarist economist Milton Friedman and Anna Schwartz. Barclay said he had never met Labour's leader, Ed Miliband, or the deputy prime minister, Nick Clegg.