Gay immigrants are mounting a campaign to demand that they be included in any unilateral moves President Obama makes this year to halt deportations, arguing that they face unusual circumstances that otherwise might leave them on the outside.

The demands pose a tricky test for Mr. Obama, who has made major inroads with gay rights groups during his time in the White House, but who will have to draw lines somewhere as he seeks to craft a more lenient deportation policy without canceling deportations altogether.