Support for culture secretary Maria Miller appeared to be ebbing as Tory MPs said David Cameron should sack or demote her in an expected summer reshuffle over her handling of an expenses dispute.

Senior members of the 1922 committee of Tory backbenchers, who have the ear of the prime minister, said they expected Miller to be removed from her post to end a barrage of criticism from Conservative-supporting newspapers in the runup to next year's general election. One MP high in the hierarchy said of Miller, who made a curt 30-second apology to the House of Commons after she was ordered to pay back £5,800 in overclaimed allowances "I think it was very unwise to give the short, perfunctory apology she did. I suspect she may be sacked, demoted or moved in June."

An unnamed senior minister said on Saturday night that Miller should resign, while a poll for one Sunday newspaper suggested more than three-quarters of voters wanted her sacked. Another member of the 1922 committee executive said he shared the view that Miller would have to resign.

Sir Ian Kennedy, chairman of the Independent Parliamentary Standards Authority, weighed in, saying MPs can no longer be trusted on expenses. He told the Sunday Times: "MPs marking their own homework always ends in scandal. It happened with expenses. It will happen with standards investigations too. Ipsa has shown that independent regulation of parliamentary behaviour can work. Our reforms have cleaned up the system."

Cameron is expected to reshuffle his cabinet soon after May's European parliament elections, putting in place the team that will take the party into the general election. On Saturday fellow cabinet ministers, including party chairman Grant Shapps, tried to bolster Miller's position, saying the MP for Basingstoke had been cleared on the substantive charges, had apologised for errors, and that it was time to "draw a line" under the matter.

But as deputy prime minister Nick Clegg failed to offer strong backing – he said Miller's future was a matter for the prime minister – and Labour raised questions over allegations that No 10 and one of Miller's aides had tried to intimidate journalists planning to write about her expenses, her future looked increasingly doubtful.

On Thursday, the commissioner for parliamentary standards, Kathryn Hudson, recommended that Miller repay £45,000 in expenses for a house she shared with her parents and had wrongly claimed was her "second home". The rules say second homes must be used exclusively for parliamentary purposes that would not include using them to accommodate parents. The cross-party House of Commons standards committee, which adjudicated on the report, overruled the watchdog and decided she only needed to hand back £5,800 in overclaimed mortgage interest. Miller was ordered to apologise to the Commons for failing to co-operate fully with the inquiries, which she did in a statement whose brevity shocked MPs.

The controversy deepened after the Daily Telegraph released an audio tape of a 2012 phone call between one of Miller's adviser's, Jo Hindley, and a Telegraph reporter, in which Hindley drew attention to her boss's role in policy on press standards – a move seen by the paper's then editor, Tony Gallagher, as a threat. No 10 and Hindley strongly deny they threatened the newspaper.

Evidence of strong-arm tactics has also come to light in emails released by the standards committee, which showed how Miller tried to persuade Hudson that the investigation into her expenses was not a reasonable one. In one, she wrote: "As should by now be obvious, a decision to uphold the complaint would be irrational, perverse and … unreasonable – that is to say it would be a decision that no reasonable decision-maker could properly reach."

Asked on Radio 4's Any Questions whether she should quit, Tory MP Jesse Norman failed to offer a ringing endorsement. He said: "I think that is a matter for her and the prime minister."

Conservative backbenchers, while sympathetic with Miller over details of her expenses, were shocked by her failure to show real contrition in the Commons and fear the dispute will continue with newspapers opposed to apparent government attempts to give politicians a role in press regulation.

The former chairman of the committee on standards in public life, Sir Alistair Graham, said he was shocked at Miller's behaviour. "The degree of lack of co-operation and the attempt to divert the commissioner from addressing the issues concerned both seem fairly exceptional.

"I think particularly for a senior cabinet minister, who you expect to show a leadership role in co-operating with whatever expenses system is around, it is pretty shocking.

"I think the public will be very shocked that the committee did overturn one of the key recommendations about how much should be repaid back, when there is a real possibility that the minister made a capital gain with the help of public funds."

Kennedy, the Ipsa chairman, said: "We have made great progress in cleaning up the problems of the past. To avoid further damage to parliament in the future, it should have the confidence to give away powers in regulating itself and see that independent regulation is the best, most transparent way forward.

MPs should "learn a lesson" from the independent system of regulation of expenses that was introduced in the way of the 2009 scandal and ensure that Ms Hudson too is "given the freedom to carry out her work and not have her wings clipped by MPs", said Sir Ian.

Labour stopped short of calling for Miller to resign but said many questions remained unanswered. It is demanding to know more about allegations she and No 10 tried to push the Daily Telegraph off the story by reminding journalists of her responsibility for policy on the press. Shadow Commons leader Angela Eagle said: "The latest revelations and the release of a recording raise further serious questions for Maria Miller and David Cameron. They urgently need to make clear what they knew about these calls and what action they took about them.

"There is also the important question of whether there has been a breach of the code of conduct for special advisers or the ministerial code. It is time for David Cameron to show some leadership, stand up and say that what has happened was wrong and to answer these questions in full."

Labour MP John Mann, who raised the first complaint over Miller's expenses with the commissioner and who has said she should be sacked, said he was now looking at whether she owed tax on expenses that had been overpaid. He added that it was "very disturbing" that the Commons standards committee had overlooked evidence that she tried to intimidate the commissioner.