"It's a good thing that for an ... immigration bill to pass, it's gotta pass the House. This is the more difficult hurdle, so let's start there," Johnson said in an interview on radio station 1130 WISN. "It could've guided the Senate's actions."

Senate Majority Leader Harry ReidHarry Mason ReidDems walk tightrope on Pompeo nomination The Memo: Teens rankle the right with gun activism Dems to party: Go on offense with Trump’s alleged affairs MORE (D-Nev.) hopes to pass the bipartisan immigration reform bill, introduced by four Senate Democrats and Senate four Republicans, before his chamber goes on recess for the July 4 holiday. 

Most recently, the Senate's debate on the bill has centered around border security. Sen. Marco RubioMarco Antonio RubioStudents gather outside White House after walkout to protest gun violence Overnight Energy: Senate confirms Bridenstine as NASA chief | Watchdog probes Pruitt’s use of security detail | Emails shine light on EPA science policy changes Senate confirms Trump’s pick to lead NASA MORE (R-Fla.), one of the eight senators pushing the proposal, has stressed that the bill can only pass if border security measures are strengthened.

"I think it's an excellent starting point, and I think 95, 96 percent of the bill is in perfect shape and ready to go," Rubio said Sunday on ABC's "This Week." "But there are elements that need to be improved. I think the debate now is about what that border security provision looks like. And if we do that, this bill will have strong bipartisan support."

But Johnson argued the immigration bill is unlikely to get the necessary border security changes it needs because it started in the Senate.

"But that's not where we're at and I don't think the Senate's going to be an effective place to actually strengthen the current bill from the standpoint of border security because I think it's pretty well cooked in," Johnson continued in the radio interview on Monday.