A bipartisan group of senators has agreed on a set of principles for a sweeping overhaul of the immigration system, including a pathway to American citizenship for 11 million illegal immigrants that would hinge on progress in securing the borders and ensuring that foreigners leave the country when their visas expire.

The senators were able to reach a deal by incorporating the Democrats’ insistence on a single comprehensive bill that would not deny eventual citizenship to illegal immigrants, with Republican demands that strong border and interior enforcement had to be clearly in place before Congress could consider legal status for illegal immigrants.

Their blueprint, set to be unveiled on Monday, will allow them to stake out their position one day before President Obama outlines his immigration proposals in a speech on Tuesday in Las Vegas, in the opening moves of what lawmakers expect will be a protracted and contentious debate in Congress this year.

Lawmakers said they were optimistic that the political mood had changed since a similar effort collapsed in acrimony in 2010. Senator John McCain, Republican of Arizona and one of the negotiators, said he saw “a new appreciation” among Republicans of the need for an overhaul.

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