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Senate Democrats are set to unveil a $2.7 billion border funding bill, far short of the president's $3.7 billion request for emergency money to alleviate the crisis of unaccompanied minors traveling across the southern border.

Appropriations Committee Chairwoman Barbara Mikulski, D-Md., presented the bill details to her colleagues at the weekly policy lunch Tuesday afternoon. Senators emerged from the meeting citing the $2.7 billion figure, a number confirmed by leadership aides.

Senate Majority Leader Harry Reid told reporters that Mikulski would unveil the bill's details later Tuesday. It's expected that the chamber will consider the measure late this week or early next week.

Emerging as a key issue in the debate over the bill: Whether to change the 2008 anti-trafficking law that makes it harder to deport unaccompanied children who come to the U.S. from other countries (excluding Mexico and Canada). Senate Republicans -- and the Obama administration -- have indicated they're interested in changing the law to make it easier to send the children back.

But some Senate Democrats have balked at that idea. Reid suggested that the border funding bill would not include changes to the trafficking law, telling reporters at the Capitol that he believes the administration has sufficient authority to make such a policy change without Congress.