David Cameron was under increasing pressure last night after he was forced to admit that the Conservative Party's millionaire backers had enjoyed private dinners in Downing Street.

Labour will try to force a minister to come to the Commons today to explain why the Tory co-treasurer, Peter Cruddas, was recorded boasting of the access a donor could buy for £250,000.

It was confirmed that Mr Cameron and his wife hosted a "small number" of dinners in his taxpayer-funded Downing Street home. Guests included Michael Spencer, the international trader and former Tory treasurer. Police are also looking at whether any law on political donations has been broken.

Mr Cruddas resigned as soon as the scandal broke, and his remarks were disowned by senior Tories, who dismissed him as a newcomer to politics who did not understand the system and was making boastful promises that he would never have been allowed to fulfil. Mr Cameron told the BBC: "What happened is completely unacceptable. This is not the way that we raise money in the Conservative Party. It shouldn't have happened. It's quite right that Peter Cruddas has resigned. I will make sure there is a proper party inquiry to make sure this can't happen again."

But Labour politicians refused to accept that Mr Cruddas, who has himself given hundreds of thousands of pounds to the Tories, was genuinely ignorant about the relationship between political influence and party donations.

Mr Cruddas, the founder of an online trading company, Currency Management Consultants, was recorded by undercover journalists from The Sunday Times promising that a £250,000 donation would be rewarded by dinners with David Cameron and Chancellor George Osborne.

He claimed that some donors had dined privately in 10 Downing Street with David and Samantha Cameron, and said issues raised by "premier league" donors would be passed to "the policy committee". It is assumed he meant the No 10 Policy Unit.

The Labour leader, Ed Miliband, claimed that the remarks raised issues too serious to be handled by an internal Tory party investigation. "There needs to be a proper independent investigation into what influence was sought, what influence was gained and what impact it had," he said.

"The independent investigation should look at what happened at the Downing Street policy unit, because apparently offers were made or cash was paid so that the people donating money would be able to have access to it. I think people are bound to ask questions about whether policy is being made in the national interest or the Conservative Party's interest. That's why these allegations are so serious."

Last night Lord Fink, who returns as the principal treasurer for the Tories, said: "Since I became the co-treasurer of the party in 2008, there has been absolutely no question of donors being able to use their donations to influence policy or to gain improper or special access to senior politicians.

"Peter Cruddas' comments are in my view entirely without basis in fact and the approach to fundraising they imply is wholly improper. It is not possible in any way to buy privileged access to ministers."