(CNN) Senate Majority Leader Mitch McConnell said Tuesday he doesn't know if President Donald Trump will sign the border security deal that would avert a partial government shutdown at the end of this week, but the Kentucky Republican and other lawmakers from both parties are urging the President to agree to it.

"I hope he'll decide to sign it," McConnell told reporters Tuesday. "We're all quite interested in that as you can imagine."

CNN asked McConnell if he has gotten any assurances from Trump that he'll get behind the deal, which does not include the more than $5 billion the President has demanded for a border wall, and McConnell said he has not but wanted Trump to wait until all the details were on paper.

Senate Appropriations Committee Chairman Richard Shelby said he would be speaking to Trump later Tuesday about the agreement.

"We believe that the caucus will support this," the Alabama Republican said. "I believe that the House will support this. I hope the President will support it."

Shelby, who has been one of the principle negotiators and who announced the agreement Monday night, defended the deal that was struck, pointing to the 55 "new miles" of border barriers in the agreement.

"The President's people were all aware and involved with us," he said. "Any legislative solution is not perfect. It's not perfect, even from my standpoint."

'Shutdowns are bad, period'

Following the longest shutdown in US history, Republicans as well as Democrats sought to pressure the President to prevent another one. Asked if the GOP should accept a deal, Republican Sen. Johnny Isakson of Georgia said "Shutdowns are bad, period."

Republican Sen. Susan Collins of Maine told reporters that "absolutely" Trump should sign the proposal -- and she renewed her concern about the option for the President to declare a national emergency to build Trump's wall, saying there's no grounds for doing so.

Funding for 25% of the federal government runs out Friday, and negotiators have been hammering out a deal for weeks that addresses funding for the Department of Homeland Security and how to address Trump's campaign promise of a border wall. After stalling out over the weekend, negotiators announced a deal last night that included $1.375 billion for barrier funding that will cover roughly 55 miles of new barrier, though there are restrictions on what exactly that money can be used for.

Sen. John Cornyn, a Texas Republican, said he flew back to Washington on Monday night with the President on Air Force One following a rally in El Paso.

"He is not ecstatic over the deal but I do think he's reviewing his options," Cornyn said. "I flew back with him last night, he didn't make any declarative statements about what his intentions are but obviously it's a compromise, and with compromises everybody's a little bit unhappy."

'I don't think he was probably going to be happy with it'

Sen. Lindsey Graham, a Republican from South Carolina who is a leading voice in his party on the immigration issue, said he needed to study the bill before knowing if he would encourage Trump to sign it. But Graham said he is concerned that the number of detention beds funded in the proposal is maintained at the current level of 40,520.

"I just think if you're limiting the bed space, in reality, that's a bad deal because we need more bed space not less," Graham said, acknowledging he needs to better understand how the policy is intended to work.

Conservative Republican Sen. Kevin Cramer of North Dakota says the deal "is not enough" on its own, but he's hoping Trump will sign it and then take executive action.

"I'd like to know what other plans he has beyond that," Cramer said. "I am still waiting to learn more."

Arkansas Republican Sen. Tom Cotton, a Trump ally who has vehemently opposed past immigration reform efforts, repeatedly said "no comment" when asked about the bill or if he would stall its passage on the floor through a filibuster.

"I'd love to see more for security," said Sen. Joni Ernst, a Republican of Iowa who is also a member of GOP leadership, but she added, "I hope we can get through this."

Sen. John Thune, a South Dakota Republican and a member of leadership, had doubts Trump would like the deal.

"I don't think he was probably going to be happy with it," Thune said.

Democrats aren't happy either

As rank-and-file members learned about the details of the conference committee's agreement in principle, no one seemed especially ecstatic. Democratic Sen. Chris Murphy said the deal is far from perfect, but that "I'm willing to swallow some things I don't like in order to make sure we don't have another catastrophic shutdown."

He added, however, that Trump could have had this deal back in December.

"We settled for a number that has been on the table for two months," Murphy said. "This job is absolutely maddening. We are spending all of our time simply trying to avert disaster and maintain the status quo. I don't know why anybody would want to run for Congress if all you do is try to convince the President to not hurt the country."

"If this is where we end up, it makes the shutdown look even stupider and even more malevolent," Murphy said.

Senate Minority Leader Chuck Schumer said, "Months of shutdown politics must come to an end" with this agreement.

"We now have a bipartisan proposal to accomplish our goals to better secure the border and avoid another senseless government shutdown. I don't know the details, but the parameters are good," said Schumer, a Democrat from New York. "We must not have a rerun of what happened a few months back where legislators, Democrats and Republicans, House and Senate, agreed and President Trump pulled the rug out from under the agreement and caused a shutdown."

To get the deal, McConnell noted that Democrats backed off their "extreme position" on the number of detention beds -- a key sticking point that scuttled negotiations over the weekend -- and the Kentucky Republican praised the proposal because it would fund "miles of new border barriers."

Will Trump sign it?

The bill must now pass both chambers of Congress and Trump must sign it in order to avert a shutdown. Whether Trump will accept the deal -- that has far less than the more than $5 billion he demanded for the wall, which led to the last shutdown -- is a key question. The White House said earlier Tuesday that Trump is continuing to weigh his options -- which still include taking executive action to secure funding for a wall.

Trump told reporters a government shutdown is unlikely at the end of the week, even if a funding agreement isn't fully satisfactory.

"I don't think you're going to see a shutdown," Trump said. "If you did have it, it's the Democrats' fault."

A senior GOP aide involved in spending talks said there was some expectation the President would vent about the deal and noted he didn't say he wouldn't sign the agreement.

The aide added: "It's not like there's a Plan B right now."

Later Tuesday, the President tweeted an even more positive message:

"Was just presented the concept and parameters of the Border Security Deal by hard working Senator Richard Shelby. Looking over all aspects knowing that this will be hooked up with lots of money from other sources.... Will be getting almost $23 BILLION for Border Security. Regardless of Wall money, it is being built as we speak!"

But according to a source familiar with this reaction, Trump wasn't completely sold by Shelby's phone call Tuesday evening on the border security compromise.

Shelby told the President that the more than $1 billion included in the deal is in the range of what the President wanted -- a hard sell considering Trump shut parts of the government down in December while demanding $5.7 billion for the wall. Trump's tweet makes clear that Shelby pitched the funding overall, but he's not yet totally convinced.

As a result, not a single White House official was willing to go on the record and say that the President will sign this compromise -- but they're hopeful. Several officials said they feel confident there won't be a shutdown, and that the number one idea floating around the West Wing right now is the President signing the compromise and then using his executive powers to secure further funding.

This story has been updated with additional developments Tuesday.