Sen. Dick DurbinRichard (Dick) Joseph DurbinLive coverage: FBI chief, Justice IG testify on critical report Hugh Hewitt to Trump: 'It is 100 percent wrong to separate border-crossing families' Opioid treatment plans must include a trauma-informed approach MORE (D-Ill.) said Sunday that the comprehensive immigration bill he negotiated with a bipartisan group of seven other senators directly addresses some of the security flaws that may have been exploited by a foreign student accused of helping bombing suspect Dzhokhar Tsarnaev dispose of evidence.

"It's hard to believe 12 years after 9/11 we're having this conversation, but you put your finger on it. There's not enough coordination between these different agencies so that we know someone should not have been readmitted to the United States," Durbin told CNN. "But our bill addresses that directly."

The Department of Homeland Security announced over the weekend that border security agents must now check that foreign students entering the country have a valid visa. Azamat Tazhayakov, who was arrested last week and accused of removing a backpack containing fireworks from Tsarnaev's apartment, did not have valid documentation when he returned to the U.S. in January, but was waved through by border agents who did not have access to the database from Homeland Security.

Other lawmakers have questioned the border security system after it was revealed that a trip to Russia by Dzhokhar's brother, Tamerlan, went undetected because of a spelling mistake.

Durbin said Sunday that the immigration reform bill "starts to do things that should have been done long ago."

Under the bill, Durbin says, the federal government would be able to track all immigrants and "know not only when they arrive, but when they leave."

The Senate Judiciary Committee will begin debate on the bill this week, but some lawmakers have already expressed concern about the legislation's long-term prospects. Sen. Marco RubioMarco Antonio RubioDem scores upset over Republican in Florida county commissioner race GOP senators introduce bill to prevent family separations at border McCain, Coons: Trump should withdraw controversial refugee nominee MORE (R-Fla.) has said that the bill would not pass the House without strengthening the border security triggers necessary for a pathway to citizenship, while Democrats have complained the bill does not include a provision for same-sex couples.

Durbin said he believed "of course" there were elements that could be improved, but also seemed to caution Republicans from trying to move the bill too far to the right.

"I have friends of mine, incidentally, who look at it from the viewpoint of Democrats and from the left and say there are things we'd like to see in it too," Durbin said. "But we've got to basically stick to the standard of what we've established, what we've agreed to."

Coincidentally, Durbin is co-sponsoring the amendment that would give American citizens the opportunity to sponsor a same-sex partner, but Republicans have warned such a provision could torpedo the bill. On Sunday, Durbin defended the amendment as "consistent with the position we should have marriage equality."

"If we can find a way through this to protect the basic right of an individual and still pass immigration reform, that's what I want to achieve," Durbin said.

He also said that Gang of Eight negotiators, who had agreed to oppose as a group amendments that had the potential to kill the overall immigration bill, "did not have a specific agreement among the eight of us about this particular issue."

But the Illinois lawmaker also said there was the potential that the entire debate over the inclusion of LGBT benefits in the bill might soon become moot.

"The Supreme Court has taken up [the constitutionality of the Defense of Marriage Act]. That decision on DOMA may preclude this whole conversation. They may help us reach the right place in this whole conversation," Durbin said.