Welfare secretary Iain Duncan Smith has instructed officials to "look again" at how the "bedroom tax" will affect disabled people, the BBC has been told.
Under the plans, social housing tenants' benefits will reduce if their home has one or more spare bedrooms.
But charities have told the government that couples who could not share a bedroom because of a disability would be unfairly penalised.
Mr Duncan Smith said he understood the concerns.
He said he had has "already issued an instruction" to officials at the Department of Work and Pensions saying, "let's look at this again".
On Thursday, the chief executives of seven charities - Carers UK, MS Society, Mencap, Macmillan Cancer Support, Disability Rights UK, Carers Trust and Contact a Family - wrote an open letter to Chancellor George Osborne and Mr Duncan Smith.
In it they expressed their concerns about the impact of the new policy on disabled people and families caring for disabled family members.
'Simply wrong'
Heléna Herklots, chief executive of Carers UK said the changes would hit families for whom an extra bedroom was essential.
"If you care full-time for a severely ill or disabled partner, their condition may mean a separate room for you to sleep is vital. Disabled children often cannot share with their brothers or sisters," she said.
"Hitting carers and disabled people with extra costs for this essential accommodation, or forcing them to move is simply wrong."
The charity said government figures showed the measure would affect 420,000 people. It said its analysis showed extra funding from government to provide "discretionary payments" to families hit by the policy would only protect 10% of the disabled people affected.
BBC home editor Mark Easton said Mr Duncan Smith recognised that one obstacle to changing the rules was how to identify people who might qualify for exemption.
However, if a solution was found it might be in place before the charge is introduced in April, our correspondent reports.
Under the government's so-called "size criteria", families will be assessed for the number of bedrooms they actually need.
The change affects council tenants, and those renting from housing associations, of working age who receive housing benefit.
It does not affect claimants who rent in the private sector.
Tenants in social housing will have their benefit reduced by 14% if they have a spare bedroom or 25% if they have two or more extra rooms.
BBC
