Up to 30 Conservative MPs could back a Commons amendment requiring councils in England to declare if they can house unaccompanied child refugees.
Tory backbencher Heidi Allen said she had tabled the amendment to the Children and Social Work Bill following the decision by the Government to close the Dubs child refugee scheme.
Ministers have been widely criticised for shutting the programme, named after the Labour peer Lord Dubs, after just 350 unaccompanied children were resettled, arguing councils could not afford to take the 3,000 called for by campaigners.
However Ms Allen said, while some authorities were struggling, others still had the capacity to take more.
"Some councils are absolutely bursting, they cannot take any more. They have done an amazing job. But there are those still stepping forward and we have not taken them up on their offers," she told the BBC.
"This problem won't go away.
"There are thousands of children who won't go away who are in Greece and Italy who have nowhere to go.
"Of course, we can't help them all.
"But they are turning to traffickers because legal, safe routes like Dubs have been closed to them.
"They will come to northern France and they will become our problem."
Ms Allen's amendment, which has cross-party support, would require councils to report annually on what capacity they have to resettle children, including unaccompanied refugees.
The move has the backing of a number of prominent Tories, including former education secretary Nicky Morgan, and Ms Allen said up to 30 Conservative colleagues could vote for it at the Bill's report stage later on Tuesday, potentially jeopardising the Government's Commons majority of just 16.
"I believe, as many do in the House of Commons, especially on my side around about 30, that we do need to play our part there," she said.