By Jeff Zeleny and Arlette Saenz

There's no shortage of criticism on Capitol Hill over how President Obama has handled the escalating humanitarian crisis at the border, but don't look for Congress to simply rubber stamp his $3.7 billion emergency spending request.

The concerns are not only coming from Republicans. Democrats have as many questions of their own over the White House proposal.

"It's something we have to do. It's something we have to do," Senate Majority Leader Harry Reid told reporters today. "How we get there, I really don't know at this stage."

READ: Details of Obama's Border Funding Request

Some Republicans are uncertain they can support a bill-even one to strengthen border security-if the costs aren't offset by budget cuts elsewhere.

"He's asking for a blank check, in essence, $3.7 billion, but no reform," Sen. John Cornyn, R-Texas, told reporters.

Many Democrats are furious at the administration's sharpening rhetoric about unaccompanied minors crossing the border illegally and the plan to deport most of them. There is resistance from many lawmakers and liberal immigration groups to changing the 2008 law at the root of the current crisis.

READ: Obama Warns Central Americans: 'Do Not Send Your Children To The Borders'

Inside the closed-door Senate policy lunches today, a heated discussion broke out among Democrats. Sen. Robert Menendez of New Jersey and other Democrats are strongly opposed to administration efforts to change the law and make it easier to send children back to Honduras, Guatemala and other Central American countries.

But Reid said he hoped to keep the bill as close to the White House request as he could, without adding other measures.

"I think we should try to stick with what we have," Reid said at his weekly news conference.

WATCH: The Invisible Crisis: Small Children Crossing the US Border on Their Own

The Senate Appropriations Committee is scheduled to hold a hearing on the emergency funding request Thursday. The House Appropriations Committee is also studying the proposal, but it remains an open question whether the House or Senate will move forward on the bill first.

It quickly became clear that swift Congressional approval was not likely. Congressional leaders said they hoped to vote on the request before the end of the month.

"We need to take a look at what he's asking for here and see if it's an appropriate response to the crisis that we have," Senate Minority Leader Mitch McConnell said.

As the president is poised to visit Texas on Wednesday, Republicans blasted him for not scheduling a tour of the border.

"To his credit, President Obama has come to Texas in the last few years to help us mourn tragedies in places like Fort Hood and West, which makes it all that much more inexplicable why we would refuse to carve out just one hour out of his fundraising schedule over the next two days in Texas to come to the border at the site of what he calls a humanitarian crisis," Cornyn said. "He really needs to go there and see this for himself."

READ: DHS Chief Blames 'Horrible' Conditions for Child Immigrant Spike, Not Deferred Action