Theresa May has questioned the “legality, practicality and efficacy” of ministers’ plan to send asylum seekers to Rwanda amid demands from MPs that the government should disclose the cost of the scheme.

The former prime minister, who devised the “hostile environment” policy, warned on Tuesday that the plan to fly those who arrive by unauthorised means to the African country could lead to an increase in the trafficking of women and children.

May made the claim as Priti Patel, the home secretary, came under repeated demands in the Commons to explain how much the scheme, announced last week while parliament was in recess, would cost.

The UK government has said it will pay an initial £120m to the Rwandan government, but will have to pay additional costs for housing, food and travel.

May, who has been blamed in part for the Windrush scandal while home secretary, questioned whether the policy was within the law. “I do not support the removal to Rwanda policy on the grounds of legality, practicality and efficacy,” she said.

The former Conservative party leader asked whether reports were correct in claiming that families would not be broken up by the government as it sought to send people to Rwanda.

“Where is her evidence that this will not simply lead to an increase in the trafficking of women and children?” May said.

Patel did not clarify whether women and children would be sent to Rwanda, but insisted the policy was legal and was needed to tackle smuggling gangs who “effectively exploit various loopholes in our existing laws”.

In response to earlier criticism of the proposal from Justin Welby, the archbishop of Canterbury, and the former archbishop Rowan Williams, Boris Johnson reportedly criticised “senior members of the clergy” during a meeting of the 1922 Committee of backbench Tory MPs on Tuesday evening.

The prime minister said they “had been less vociferous in their condemnation on Easter Sunday of Putin than they were on our policy of illegal immigrants”, according to PA Media.

Andrew Mitchell, the former Tory international development secretary, was one of a series of backbenchers who joined opposition MPs in expressing concerns at the policy, saying it “will simply not work”.

Patel refused to directly answer MPs’ questions on whether there was a cap on the cost of each person forcibly deported, on top of the £120m economic deal with Rwanda.

Yvette Cooper, the shadow home secretary, told MPs that the government was looking for help from Rwanda because Home Office decision-making had “totally collapsed”.

“On [Patel’s] watch, the Home Office are only taking 14,000 initial asylum decisions a year. That is half what they were doing five years ago.

“So, the costs of the UK taxpayer have soared by hundreds of millions of pounds because she isn’t capable of taking the basic asylum decisions. And because she’s not capable of taking those decisions, she is trying to pay Rwanda to take those decisions instead,” she said.

The Scottish National party’s home affairs spokesperson, Stuart McDonald, said Patel’s policy would not hurt smugglers but would harm people who had fled persecution.

“Her government has completely failed to stop abuses in UK detention centres, never mind centres that are 5,000 miles away.

“In short, this disastrous policy has nothing to do with the global migration crisis, it is everything to do with distracting from the PM’s political crises,” he told MPs.

Patel said the final comment was “absolutely unacceptable” and accused opposition MPs of “playing party political games”. She urged the House of Lords, which has rejected the nationality and borders bill twice, to accept it.

“Once again [I] urge the ‘other place’ to follow this elected House in backing the bill,” she said.

Labour’s Dame Diana Johnson, who chairs the home affairs select committee, asked: “Who would actually be eligible to be sent to Rwanda? Is it going to be single young men, is it going to be women and children?”

Patel replied: “Everyone considered for relocation will be screened, interviewed and have the right access to legal advice and services.”

Patel told the Commons that a reported objection from Matthew Rycroft, her permanent secretary, was not evidence that he opposed her plans.

“He stated, in his role as accounting officer, that the policy is regular, proper and feasible, but that there is not currently sufficient evidence to demonstrate value for money,” she claimed.

In an exchange of letters with Patel, Rycroft wrote: “I do not believe sufficient evidence can be obtained to demonstrate that the policy will have a deterrent effect significant enough to make the policy value for money.”

Last year, more than 28,000 migrants and refugees made the crossing from mainland Europe to Britain, a fraction of the number arriving in other European countries.