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(CNN) —  

Time is ticking and there is still no guarantee that the House of Representatives and Senate can resolve their differences to pass by the end of the week a package sending an additional $4.6 billion in aid to the growing crisis at the US southern border.

As for overhauling the country’s asylum laws, as President Donald Trump appeared to request last week: Good luck.

But first a reality check on Trump’s ask: Last week, Republican Sen. Lindsey Graham of South Carolina was all set to mark up his own asylum bill that would have required migrants to apply for asylum in their home countries or in Mexico, increased the amount of time migrant children be kept in custody from 20 days to 100 days and eased restrictions for officials to deport unaccompanied minors back to Central America. But, Graham called off the committee vote after Sen. Dick Durbin, the Senate’s Democratic whip and Graham’s longtime immigration partner, asked Graham to work on something together.

Graham’s bill would have passed out of committee but only along party lines. Durbin argued there is a chance to do something bipartisan, but the obstacles are many. Democrats don’t want to touch the Flores settlement – which requires the government to release children from immigration detention without unnecessary delay as well as giving detained minors a certain quality of life – that has been at the crux of many of the complaints about overcrowding at family shelters.

So far, it doesn’t appear they have really reached an agreement or have started negotiating in a substantial way. The men met last week with the President’s son-in-law and senior aide Jared Kushner. But, it is very unlikely a proposal would be ready, agreed to and passed this week before the recess.

Barring some radical breakthrough, it’s hard to imagine policy changes being attached to the moving border supplemental.

What is the $4.6 billion for?

If Democrats and Republicans cannot agree on a border supplemental, a key department within Health and Human Services – the Office of Refugees and Resettlement – runs out of money at the end of the month. The ORR is critical because it operates shelters for unaccompanied migrant kids.

There have also been a stream of reports – including from CNN – of subpar conditions for children and families in CBP’s care. And everyone – Republicans and Democrats, liberals and conservatives on the Hill – agree there is a crisis on the border that has to be dealt with.

Last week, the Senate Appropriations Committee passed out of committee a bipartisan $4.6 billion border supplemental package. The bill included money for the Department of Defense, Justice, Immigration and Customs Enforcement and even FEMA to deal with the crisis on the border. The bill also included $2.88 billion to shore up funding of the Office of Refugees and Resettlement that handles the housing of unaccompanied migrant kids. The bill passed 30-1 in committee.

What to expect in the House Tuesday

The House will likely vote on their own version of the border supplemental that differs from the Senate’s in key ways including not providing money to the Department of Defense and providing more restrictions on how the money can be used.

What will the Senate do? It’s unclear when exactly the upper chamber will vote. The National Defense Authorization Act is on the floor, but if the Senate got consent, they could move the border supplemental quickly. Everything is possible with consent in the Senate.

So what happens next?

A few scenarios could play out.

  1. The House passes their bill. The Senate passes their bill. Then, the Senate forces the House to take up their bill arguing theirs is bipartisan and passed 30-1 out of committee (this is what GOP aides argue behind the scenes makes the most sense).
  2. The House passes their bill. The Senate passes their bill. Minor changes are made to amend the Senate bill so it is somewhat closer to the House bill and both chambers pass a new bill (this option is tough because there is limited time and NDAA Is on the Senate floor this week).
  3. The House passes their bill. The Senate passes their bill. There is an impasse, and nothing passes.

CNN’s Ted Barrett, Priscilla Alvarez, Elizabeth Elkin and Emily Smith contributed to this report.