Mr Straw told concerned MPs that he intended to increase the cash allowance for asylum seekers to make them less dependent on a voucher scheme which has been attacked as "humiliating".
Under the new arrangements, a family will be allowed pounds 40 cash a week, instead of pounds 21, as a proportion of their pounds 90.80 income support. The remainder will be paid in vouchers.
Some MPs have voiced concerns that the voucher system imposes too great restrictions on asylum seekers, particularly on parents who are unable to buy items for their children.
In further concessions to opponents, Mr Straw promised to recruit around 300 extra immigration officials to improve the speed and efficiency of the system. He pledged to prioritise family cases by dealing with them within six months from next April.
But he declined to take families - who comprise 13% of all asylum applications - out of the system altogether, as some Labour MPs had hoped.
One Home Office source said: "People are understandably concerned about families. We recognise the point and therefore the focus of the changes have been to bolster the support for families and to speed up the system. But we could not take families outside the system."
The concessions are unlikely to placate all the dissenters.
Neil Gerrard MP, chair of the All-Parliamentary Group on Refugees, has gathered 35 signatures supporting an amendment to the bill to take families out of the voucher system altogether.
The bill has its report stage in Parliament next week.
Last night he said: "I still feel there will be quite a number of people who are unhappy and will say this is not enough."
The government is worried about a repeat of the rebellion it faced last month when 67 MPs defied its plans to cut incapacity benefit for the disabled. Mr Straw will reveal details of the concessions today in a speech to the Parliamentary Labour Party.
Cash allowances for adult asylum seekers will increase from pounds 7 a week to pounds 10 a week, while children will also be allowed pounds 10 a week instead of pounds 3.50.
The benefits amount to about 70% of income support.Reuse content