Mr Trump is demanding $5 billion in the budget but Democrats are expected to offer no more than $1.3 billion, to fund security measures such as surveillance technology and personnel.
For the past two years, Pelosi has repeatedly said she had initially meant to retire after the 2016 elections but chose to stay on and continue leading House Democrats after Trump's upset victory.
Tweeting ahead of the White House meeting, Trump claimed that the wall was needed to prevent "large scale crime and disease" brought by illegal immigrants.
Schumer said Wednesday it is "nearly impossible" to negotiate with Trump, accusing the president of peddling "blatant and unsafe falsehoods" about the wall, including his widely refuted claim that Mexico will pay for it.
Mr Schumer and other Democrats supported a 2006 law that authorised hundreds of miles of fencing along the southern border, McConnell said, urging Democrats to again back physical barriers - by whatever name - along the border.
McConnell warned against a government shutdown following Trump's meeting with Pelosi and Schumer.
In a poll related on Tuesday by NPR/PBS NewsHour and Marist Poll, 57 percent of Americans said Trump should compromise on the wall to avoid a government shutdown.
Moments after releasing a statement saying she was "comfortable with the proposal", seven of her critics distributed their own joint comment, pledging to back the 78-year-old Pelosi.
At her press conference on Thursday, Pelosi also addressed Democratic desire to inspect Mr. Trump's tax returns.
"Chuck and Nancy must give us the votes to get additional Border Security!" he wrote, referring to Schumer and Pelosi. "I understand it was a rather spirited meeting", McConnell told reporters when asked by about Trump's threat.
Trump has been threatening for months to partially close government unless he gets more money for the wall than Democrats have been willing to give him.
The agreement stipulates the top three House Democrats will be allowed to keep their posts for three terms, with the option of a fourth term if they win two-thirds support from the Democratic caucus.
Trump wants a funding package for the next fiscal year to include at least $5 billion for his proposed border wall with Mexico, an idea Democrats have flatly rejected.
If Trump rejects that, Democrats would likely urge a continuing resolution that funds all the remaining appropriations bills at current levels through September 30, said an aide who was not authorized to discuss strategy by name and requested anonymity. While Trump once promised to build a wall "and have Mexico pay for it", he has been limited to making minor upgrades to the existing fence since.