Shabana Mahmood has been urged to “read the room” on Labour opposition to her immigration policy, as No 10 declined to rule out watering down her plans to make foreign workers wait 15 years to get permanent residency.

Mahmood, the home secretary, is consulting on plans to increase the period before foreign workers can claim indefinite leave to remain from five to 15 years, backdated to apply to anyone arriving from 2021 onwards.

She has previously hinted that her plan to make it harder to claim permanent residence could be watered down by exempting care workers from the new rule.

However, there were reports on Tuesday that she was holding firm against making changes. A senior ally told the Times that she had the prime minister’s support, saying: “Andy has been on Shabana’s side throughout this row over ILR. He is holding the line.”

There is already a cabinet split over the policy, which deepened on Wednesday when Miatta Fahnbulleh, the energy secretary, repeatedly refused to back the specific plans to make foreign workers wait 15 years to claim permanent residency in the UK.

This week it was reported that Fahnbulleh appeared to criticise the plans on her Peckham constituency website in a post about a community discussion that was later removed. She was quoted as saying: “If we continually prolong the time people have to wait for settlement, we can also prolong the process of building cohesion.”

This year the housing secretary, Angela Rayner, publicly criticised the plans, and dozens of Labour MPs oppose the changes.

One of the leading opponents, Mike Tapp, a former immigration minister who clashed with Mahmood over the policy, said the home secretary should “read the room” in relation to the strength of feeling about the changes.

“It’s important we do the right thing. If people have come to the country and genuinely worked in the sector, they shouldn’t be punished and should have the right to apply for settlement after five years as they expected,” he said.

MPs campaigning for changes to Mahmood’s policies believe they have a chance of convincing Andy Burnham to exempt care workers, given reforming adult social care is an issue close to his heart. However, the Home Office has said that would lead to an exodus of foreign workers from the sector when they are no longer bound by the terms of their visa.

No 10 stressed on Wednesday that Mahmood was “carefully considering” more than 200,000 responses to a consultation before finalising the reforms.

Asked whether tweaks to the home secretary’s revamp – including no longer applying the policy retrospectively – could be ruled out, Burnham’s press secretary said: “We set out proposals for consultation. We’ve consulted on what the transition arrangement should look like for people already in the UK who have not received settled status.

“The home secretary is leading a thorough piece of work to ensure we get this right and have the right arrangements in place.”

In her interview with Sky News on Wednesday, Fahbulleh repeatedly avoided the question on whether she would back an increase in the waiting period for permanent residency to 15 years. She said: “We have got to reform our immigration system, and my view is that we need a system that is based on the principle of compassion, contribution. And so there is a package of reforms. The home secretary has been consulting on this. She’s looking at the range of responses, and she will come out with the measures that we take forward.

“No one can look at our immigration system – and I say this as someone that came to this country as a refugee – and not believe that it’s broken and not believe that we need to fundamentally reform it.

“But our principles has to be a system where we’ve got control, but ultimately a system that when people do come to this country, they are able to integrate, they are able to contribute, and that we both respect but also value that contribution.”

She said she supported “reforms that say that people have to contribute and make a contribution to this country in order to get citizenship”, adding: “I 100% back Shabana Mahmood and everything she is trying to do.”

But she avoided multiple questions on the prospect of a 15-year wait. When it was put to her that she did not back that specific proposal, Fahnbulleh said: “Don’t … put words in my mouth.”

A government source said that “the minister clearly supported the policy and process” in her interviews, “but while subject to review of the consultation, the final policy is not set – so ministers cannot offer opinions on specific parts of it”.

A Home Office spokesperson said: “This government has announced reforms that mean settlement must be earned and not automatic. We are currently considering responses to our consultation on elements of these reforms.”