(Reuters) - U.S. President Barack Obama is poised to give relief from deportation to millions of undocumented immigrants who are parents of U.S. citizens or of permanent legal residents, according to a source familiar with White House deliberations.

Obama has promised to lay out the details of an executive order on immigration. The action could come as early as this week.

The source, who asked not to be identified, said some details were not yet available on which parents of citizens or permanent residents would be included. The Obama administration, the source said, had been looking at options including those parents who have been living in the United States for five years or 10 years.

A top Obama aide is scheduled to have lunch with Senate Democrats on Thursday on Capitol Hill. White House chief of staff Denis McDonough, who will discuss the state of the economy and the post-election legislative agenda, is likely to be pressed on the immigration issue in the closed-door luncheon.

On Monday, in an interview with Univision, Senate Majority Leader Harry Reid, a Democrat, said that Obama should move on immigration "now."

Obama is expected to take actions to allow some undocumented people to live here at least temporarily without the threat of deportation and to hold jobs in the United States. Obama's executive order could also include further border security steps, according to sources. Obama is expected to stress that he wants to focus efforts on deportations of illegal residents with serious criminal backgrounds.

Obama repeatedly has warned that he would take steps to fix immigration problems because Republicans in Congress have refused to pass legislation. He also has said that even with his unilateral steps, Congress could still replace his measures with permanent legislation.

For the past two weeks, Republicans in Congress have been looking at ways to stop Obama from carrying out these anticipated actions, arguing that only Congress should initiate such moves through legislation.

While the Senate in 2013 passed sweeping bipartisan immigration legislation, Republicans in the House of Representatives have blocked such a bill, saying they first want to concentrate on further securing U.S. borders.