But House Speaker John A Boehner has signalled any action on immigration is unlikely this year because House Republicans do not trust President Barack Obama on the issue.
"The American people, including many of my members, don't trust that the reform that we're talking about will be implemented as it was intended to be," Boehner said yesterday, the day the poll was released.
The CNN/ORC International survey indicates that a majority of the public says that the government's main focus should be legalising the status of the undocumented rather than border security with Mexico.
According to the poll, 54 per cent say the top priority for the government in dealing with the issue of illegal immigration should be developing a plan that would allow undocumented immigrants with jobs to eventually become legal US residents.
Just over four in ten questioned say the main focus should be developing a plan for stopping the flow of undocumented immigrants into the US and for deporting those already here.
"The Republicans' insistence that border security be the primary focus of US immigration policy may have been a popular stand in 2011, but not necessarily in 2014," said CNN Polling Director Keating Holland.
"American attitudes toward undocumented immigrants have changed. Starting in 2012, most Americans have said that the government's focus should be on a plan that would allow those immigrants to become legal US residents. A majority has consistently taken that position since that time - 56 per cent in 2012, 53 per cent in 2013, and 54 percent in the current poll," he said.
The Democratic-controlled Senate last year passed a bipartisan illegal immigration bill that included an eventual pathway towards citizenship for most of the 11 million undocumented immigrants in the US.
According to the poll, more than eight in 10 support such a plan. There is little partisan divide, with 88 per cent of Democrats, 81 per cent of independents and 72 per cent of Republicans in agreement.
The poll was conducted for CNN by ORC International from January 31- February 2, with 1,010 adults nationwide questioned by telephone.