Updated: 12:01 p.m.

Senate Democrats pressed again Monday for lifting the oil spill liability cap — a move that so far has been met by staunch GOP opposition.

Senate Democratic Conference Secretary Patty Murray (Wash.) and Democratic Senatorial Campaign Committee Chairman Bob Menendez (N.J.) continued the push by calling for the approval of legislation that would make oil companies responsible for paying the entire cost of cleanup and recovery in the wake of a spill like the one currently ravaging the Gulf Coast.

“The only group of people who benefit from the bill being blocked are BP and the oil companies,” Murray said during a conference call with reporters. “We are going to keep moving forward on this; we know it’s absolutely critical.”

Both Murray and Menendez also maintained the spending cap should be increased to an unlimited level even if BP agrees to set aside $20 billion for a cleanup fund, as a majority of Senate Democrats requested over the weekend.

“While I think the $20 billion is incredibly important as a very significant down payment, [I] just don’t want to limit the liability to that,” said Menendez, who joined 53 Senate Democrats in signing a letter to BP CEO Tony Hayward calling on the oil giant to create an independently administered $20 billion account to pay for cleanup costs in the Gulf.

Republicans have blocked earlier attempts on the floor by Democrats to increase the liability cap from $75 million to $10 billion, making the case that the amount may not be appropriate. Sen. David Vitter (R-La.) has a separate proposal that would require BP to pay an unlimited amount for cleanup costs for the Gulf Coast spill. Menendez complained that Vitter’s measure “creates a unique liability for one company.”

A senior GOP aide declared it “odd” that Democrats have blocked action on Vitter’s proposal aimed at BP while seeking to push an unlimited cap for all oil companies. The aide maintained there is “bipartisan agreement” to hold BP accountable for the entire cost of cleanup in the Gulf and noted that “several of our members, including those representing coastal states, have thoughtful proposals on the issue of liability that will be offered for discussion in the coming days.”

The aftermath of the April 20 Deepwater Horizon explosion off the Gulf Coast continues to dominate on Capitol Hill. Senate Democrats will meet Thursday to discuss a path forward on energy reform, a legislative priority that has received fresh attention in the wake of the spill. Meanwhile, President Barack Obama will travel to the Gulf region again Monday for his fourth trip since the spill, this time visiting Mississippi, Alabama and Florida.