A cross-party group of MPs is confident it will be able to use an amendment to the immigration bill going through parliament to stop people being held indefinitely in immigration detention centres.

Harriet Harman, the chair of the joint committee on human rights, has persuaded the former Brexit secretary David Davis and other former Conservative ministers to back the move, which would stop thousands of foreigners facing deportation every year being held in detention for more than 28 days.

The Home Office has yet to approve the proposal but Harman, a former deputy Labour leader, said she was “very confident” her amendment would be accepted because of the strength of her case and the breadth of cross-party support.

She said new figures released to her committee by the Home Office backed the case for reform, because they disproved claims that most of those held in detention centres for more than 28 days pending deportation were in the category of “foreign national offenders” – people who have previously served at least a year in jail.

In the 1990s there were only 250 places where migrants might be held in detention before being deported, and even then these were often not full, she told the Guardian. “Now we’ve got 2,500 places, and they are bursting at the seams.”

Thousands of people every year are held for more than a month in the detention centres and in some cases people have been kept locked up for two years or more.

Harman said that in the past it was often assumed most of those detained for more than a month were foreign national offenders. But the Home Office sent figures to her committee showing that in 2017 10,331 people were detained for more than 28 days and 7,015 – about 70% – were not in the offender category.

“For every individual traumatised by indefinite detention, that’s reason to change the policy. But it is now happening on such a scale that it is really important to deal with it,” Harman said.

“The Windrush cases have shown the extent to which the Home Office is capable of acting in a completely arbitrary way. In the light of the injustices exposed by Windrush, and in the light of the fear that this could be inflicted on some of EU nationals after Brexit, it would be really quite wrong for the Home Office to insist that they strike down this amendment.”

On Monday MPs will debate the second reading of the immigration and social security coordination (EU withdrawal) bill, the legislation that will impose new immigrationcontrols on EU nationals after Brexit.

The Harman amendment, which is also backed by Dominic Grieve, the Conservative former attorney general, and Andrew Mitchell, the Conservative former international development secretary, would make it illegal for anyone to be held in a detention centre for more than 28 days unless a judge ordered a further 28-day extension.

Initially the proposed law would not cover foreign national offenders but the amendment would also allow the home secretary to extend it to cover them at a later point, by order.

Harman said the home secretary, Sajid Javid, had agreed to meet MPs backing the amendment, which will be debated at a later stage in the bill’s passage through parliament.

She said she expected him to be accommodating because he could understand the “political arithmetic” in the Commons.