Eighty-four Church of England bishops have revealed that they wrote to David Cameron last month urging him to accept at least 50,000 refugees from Syria.
The UK’s decision to accept 20,000 by 2020 was not adequate and most people wanted to offer more help, they said.
The Bishop of Durham, the Rt Rev Paul Butler, said it was “disheartening” they had not had a “substantive reply”.
Defence Secretary Michael Fallon said “nobody is doing more” than the UK “to help the refugees in their camps”.
The government has offered to accept 20,000 refugees from camps bordering Syria. It has also provided £1bn in aid to Syria, with an extra £100m given to charities to help thousands displaced by the conflict.
There has been debate in recent months about how countries should respond to the migrant crisis. This weekend Hungary closed its border with Croatia, while Slovenia put its army on standby to deal with migrants entering the country.
Bishop Butler said: “As the fighting intensifies, as the sheer scale of human misery becomes greater, the government’s response seems increasingly inadequate to meet the scale and severity of the problem.
He added: “There is an urgent and compelling moral duty to act which we as bishops are offering to facilitate alongside others from across civil society.”
Downing Street said the government wanted to tackle “the causes and consequences” of the refugee problem and that the UK was the second biggest donor in the world towards helping refugees in Syria, Lebanon, Jordan and Turkey.
Analysis: BBC religious affairs correspondent Caroline Wyatt
The decision by the 84 Church of England bishops to go public with their private letter to David Cameron is unusual, and an indication of their deep frustration at not having had what they would see as an adequate response.
But it is far from the first time that Church leaders have clashed with government. The bishops had to defend themselves from charges of political bias earlier this year, ahead of the general election, when they released an unprecedented manifesto that said it was the “duty” of every Christian to vote.
That 52-page letter warned that people felt “detached” from politics and called for a “fresh moral vision of the kind of country we want to be”. Although it was careful to praise the work of some earlier governments from both left and right, it was seen by many Conservative MPs as a distinctly left-leaning document, including references to the Trident nuclear deterrent, Britain’s relationship with the European Union and the welfare state.
Nonetheless, the bishops themselves would argue that it is their duty to offer moral leadership, and to speak out when they feel strongly that they and their flock wish the government to do more, and offer their help on the major issues of the day – however unwelcome that message may be to some.
In their letter, sent on 10 September, the bishops said they “recognise and applaud the leadership” Mr Cameron had shown when he announced the UK would accept 20,000 refugees but added the UK should do more to help tackle “one of the largest refugee crises ever recorded”.
“We believe such is this country’s great tradition of sanctuary and generosity of spirit that we could feasibly resettle at least 10,000 people a year for the next two years, rising to a minimum of 50,000 in total over the five-year period you foresaw in your announcement,” they wrote.
The letter, signed by 84 of the Church’s 108 bishops, also said they would encourage churches and congregations to make spare housing available to refugees and promote foster caring.
Bishop of Manchester David Walker, a signatory, said he had come under pressure from parishioners to encourage action.
“People want to know what we are going to do,” he told BBC Breakfast.
But Mr Fallon said the “real issue” was in Syria.
He told the BBC: “We are spending £1bn helping the refugees in the refugee camps in Syria and now we have announced that we will take 20,000 – 5,000 a year for the rest of this parliament – which is a number we think we can reasonably accommodate.”
Neither the Archbishop of Canterbury Justin Welby nor the Archbishop of York John Sentamu signed the letter. Both have called for a compassionate response to the refugee crisis.
The Most Rev Justin Welby has previously offered to help with sanctuary for refugees in the form of a four-bedroom cottage in the grounds of Lambeth Palace.
The bishops’ letter comes a week after leading former judges and lawyers criticised the “slow and narrow” response to the crisis.