Prime Minister David Cameron has denounced the party's former treasurer for boasting that a big enough donation could lead to high-level access.
He said Peter Cruddas's claims, filmed by undercover Sunday Times reporters, were "completely unacceptable". Mr Cruddas quit hours after publication.
The PM pledged a "party inquiry" into the claims that £250,000 would get donors a private dinner with him.
Labour Leader Ed Miliband called for a "proper independent investigation".
Mr Cruddas was secretly filmed saying that a donation of £250,000 gave "premier league" access to party leaders, including private dinners with Mr Cameron and Chancellor George Osborne, and with any feedback on policy shared with Downing Street.
He was heard initially saying that it was not possible to buy access to the prime minister.
But he then went on to discuss what access different size donations would get.
He was speaking to the reporters posing as staff from a fake wealth fund based in Liechtenstein who were interested in doing business in the UK.
'If you're unhappy...'
He told them: "Two hundred grand to 250 is premier league… what you would get is, when we talk about your donations the first thing we want to do is get you at the Cameron/Osborne dinners."
He said they would be able to ask Mr Cameron "practically any question you want".
"If you're unhappy about something, we will listen to you and put it into the policy committee at number 10 - we feed all feedback to the policy committee."
Mr Cameron said: "What happened was completely unacceptable. This is not the way 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."
In his resignation statement, Mr Cruddas said: "I deeply regret any impression of impropriety arising from my bluster in that conversation.
"Clearly there is no question of donors being able to influence policy or gain undue access to politicians.
"Specifically, it was categorically not the case that I could offer, or that David Cameron would consider, any access as a result of a donation. Similarly, I have never knowingly even met anyone from the Number 10 policy unit."
During a review of party funding by the independent Committee on Standards in Public Life last year, the Conservatives suggested an annual cap of £50,000.
This was dismissed by the committee as it amounted to twice the average salary and over a five-year Parliament added up to £250,000.
Instead, a cap of £10,000 was recommended by the committee, with an extra £23m of taxpayers' money to reduce reliance on "big money" donations.
Committee chairman Sir Christopher Kelly said on Sunday: "The only way to remove the suspicion surrounding very large donations would be to ban very large donations.
"That requires all the parties to address something very hard.
"For the Conservatives it means giving up their advantage of having more wealthy supporters. For Labour, the relationship with the trade unions," he said.
The Conservative Party currently has several levels of donation, with the top one being the Leader's Group,where for an annual donation of £50,000 donors can be invited to join Mr Cameronand other senior figures from the Conservative Party at dinners, post-Prime Minister's Questions lunches, drinks receptions, election result events and important campaign launches.
Mr Cruddas had been involved in fundraising for the Conservative Party since June last year, and took over as the party's principal fundraiser earlier this month.
Lord Fink will now return as principal treasurer, the party announced on Sunday morning, with Michael Farmer acting as co-treasurer.
Labour called for an independent inquiry by the Committee on Standards in Public Life.
It said it wanted the names of Tory donors who have visited government property - including Downing Street and Chequers - and of those who have made submissions to the Downing Street policy unit, to be published.
Mr Miliband told the BBC that a party inquiry into what had happened was not good enough.
"These allegations can't be swept under the carpet - there needs to be a proper independent investigation into what influence was sought, what influence was gained and what impact it had," he said.
"We need to know what happened, who paid money, what interaction there was between the prime minister and the chancellor and the people who paid money."
Mr Miliband said it "would be right" for the prime minister to make a statement to Parliament on the issue.
"I think people are bound to ask questions about whether policy is being made in the national interest or in relation to the Conservative Party interest," he said.
"This goes to the heart of the way that policy is made - it goes to the heart of Downing Street."
Labour MP Michael Dugher wrote to Mr Cameron on Sunday, saying: "Given the seriousness of the allegations about how government is conducted, it is not appropriate for the Conservative Party to investigate itself."
He said he particularly wanted to know, in light of last week's Budget, the details of all donors who had made representations, both written and orally, on changes to the 50p tax rate.
'Reform' of funding system
Coalition partners the Liberal Democrats were also concerned. Lib Dem Chief Secretary to the Treasury Danny Alexander said: "It's utterly disgraceful and there is no place for this sort of unacceptable behaviour in British politics."
He also said "reform" of funding system was necessary.
He said there was a "perception that people who make large donations - be they wealthy people from the city or trade unions - have influence. They should not have that influence, nor the perception of that influence."
Cabinet Office Minister Francis Maude said the party would continue to make clear to donors that there was "no question of people being able to influence policy through making donations".
"Parties need money and it is right that that should be strictly regulated, as it is," he said.
"Our compliance procedures are very rigorous and obviously, as David Cameron has said, we'll be looking into this very urgently to ensure that they're even more rigorous and always complied with to the full."