Dems fear that a drop-off in minority voters could severely cost them at the polls this fall.
“Women are talking about this issue," Sen. Mark Begich (D-Alaska.) said.
Senate Majority Leader Reid (D-Nev.) said his goal was to finish a bill by August recess.
The Senate leader promised to focus more attention on limiting abortions in a GOP-led Senate.
"We're at the end of the line," Sen. Robert Menendez (D-N.J.) said.
The "Tea Party-driven House" is sitting on their hands, he said.
McConnell mocked red state Dems he claimed had done little to approve the project.
Democrats are sounding the alarm about spending by outside conservative groups.
Despite opposition from House GOP, McConnell is open to moving the bill in Senate.
Democratic senators have been sidelined in legislative debates.
The possible GOP White House contenders have adopted a cautious approach on Iraq.
The bipartisan Senate plan would raise the gas tax by 12 cents over the next two years.
Graham said Republicans can get "back in the game" if they solve the immigration problem.
The GOP senator called the situation in Iraq "the greatest threat since the Cold War.”
Schumer said Cantor's defeat will make it easier to move immigration reform.
Reid credits Graham for winning primary after backing immigration reform.
The move comes amid growing concern over the massive rise in illegal young immigrants.
The Senate Armed Services Committee will hold a closed-door briefing Tuesday.