Sen. Michael BennetMichael Farrand BennetColorado senators pitch immigration compromise Colorado senators mark Olympics with Senate hallway curling GOP Senate candidate fundraising lags behind Dems in key races MORE (D-Colo.), one of the members of the "Gang of Eight" who crafted the Senate immigration bill that passed one year ago, called on the House to move on the issue.

"Tomorrow marks 365 days, one year, since the Senate acted to fix the problems and pass bipartisan immigration reform," Bennet said. "Yet here we are, still waiting for the House of Representatives to do the same."

Bennet disputed House GOP statements that Congress must enforce border security before considering a broader immigration overhaul.

"There's one border security bill that's passed the United States Congress, and that's the bill that passed the Senate," Bennet said. "If the House wants to secure the border first, as the Senate bill does, let's see their legislation."

Bennet suggested that the Senate bill could be cut into multiple separate measures if House Republicans do not want to consider one massive piece of legislation.

"I for one am willing to consider looking at this bill in smaller pieces, as long as all problems with the system are addressed," Bennet said.

Sen. Jeff SessionsJefferson (Jeff) Beauregard SessionsUnder pressure, Trump shifts blame for Russia intrusion Overnight Tech: Judge blocks AT&T request for DOJ communications | Facebook VP apologizes for tweets about Mueller probe | Tech wants Treasury to fight EU tax proposal Overnight Regulation: Trump to take steps to ban bump stocks | Trump eases rules on insurance sold outside of ObamaCare | FCC to officially rescind net neutrality Thursday | Obama EPA chief: Reg rollback won't stand MORE (R-Ala.), one of the most vocal opponents of the Senate immigration bill, argued that an immigration overhaul would make it more difficult for American citizens to find jobs in a difficult market.

"People are having a hard time finding jobs," Sessions said. "We do not need to be surging illegally the number of immigrants coming into the country." 

Another member of the Gang of Eight, Senate Majority Whip Dick Durbin (D-Ill.), similarly blasted the House for inaction on the Senate immigration bill earlier Thursday afternoon.