WASHINGTON — President Obama and his allies may soon confront a difficult decision: whether to abandon the creation of a new path to citizenship for 11 million illegal immigrants and accept tough border security and enforcement measures that they have long criticized.

Those are some of the concessions that Speaker John A. Boehner of Ohio signaled he would demand in exchange for a willingness to overhaul the immigration system. Mr. Boehner outlined those standards in a one-page document released on Thursday, and if they lead to legislation, Democrats and immigration advocates will be pressured to compromise.

Mr. Obama hinted in an interview broadcast on Friday that he was open to a plan that would initially give many undocumented workers a legal status short of citizenship, as long as they were not permanently barred from becoming citizens.

“If the speaker proposes something that says right away ‘folks aren’t being deported, families aren’t being separated, we’re able to attract top young students to provide the skills or start businesses here, and then there’s a regular process of citizenship’ — I’m not sure how wide the divide ends up being,” Mr. Obama said in an interview on CNN.