The Right To Rent pilot scheme has led to claims landlords are turning away tenants they feel will not easily pass an identity check.
They are also charging extra fees of up to £100 to carry out ID checks on tenants.
The allegations come from one of the groups monitoring how the scheme is running in the West Midlands.
The Joint Council for the Welfare of Immigrants (JCWI) is asking landlords and tenants about their experiences with Right To Rent.
Tenants say they are paying extra fees of up to £100 to landlords and letting agents for ID checks since the six-month pilot scheme started in December 2014.
If tenants do not pass the check, landlords cannot rent out a property and if they allow a tenant to move in without passing an ID check, they could face fines of up to £3,000 for offering a home to an illegal immigrant.
A JCWI spokesman said Right To Rent is stopping some British nationals from moving into a home because they cannot afford the £72.50 fee for a passport and that many home seekers from outside the European Union are facing discrimination because they have problems proving they have a right to live in the UK.
“Landlords have told us that they are unlikely to let someone view a home who might have problems with their documentation,” said the spokesman.
“This impacts on many migrants and British nationals with a right to live here who may not have a passport or birth certificate.”
“An American also complained her British husband could view homes which she had been told were unavailable.”
Since December, landlords and letting agents in Birmingham, Walsall, Sandwell, Dudley and Wolverhampton have had to check a tenant’s identity and right to reside in the UK before agreeing a tenancy agreement and giving them the keys to a home.
The checks apply to everyone aged over 18 intending to live at the property.