“I believe we can get a deal done,” House Speaker McCarthy told reporters.
While acknowledging areas of disagreement, Mr Biden said a default was “off the table”.
The debt ceiling is a spending limit set by Congress which determines how much money the government can borrow.
Failure to raise it beyond the current cap of roughly $31.4tn (£25.2tn) by June could result in the US defaulting on its debt.
That would mean the government could not borrow any more money or pay all of its bills. It would also threaten to wreak havoc on the global economy, affecting prices and mortgage rates in other countries.
Republicans, led by Mr McCarthy, have been demanding more than $4tn spending cuts in return for raising the ceiling. Democrats have refused and instead are offering to keep spending flat.
Treasury Secretary Janet Yellen has reiterated that the US will likely default on its debt as early as 1 June if no deal is reached.
The tone of the talks at the White House appeared to be more optimistic after weeks of divisive partisan discourse. But it is unclear how quickly the two sides can reach a deal.
“We don’t have an agreement yet,” Mr McCarthy said. “But I did feel the discussion was productive in areas that we have differences of opinion.”
“Biden and I will talk everyday until we get this done,” he said.
Earlier, the House speaker emphasised that a deal needed to be reached this week to give Congress adequate time to meet the 1 June deadline.
He estimated it would take about 72 hours for the agreement to be written, read and voted on.
Treasury Secretary Janet Yellen issued a warning letter to Congress on Monday maintaining that the US would likely run out of money to pay its bills as early as 1 June without a debt limit increase.
She heightened the urgency and called the possibility of a default in early June “highly likely”.
“If Congress fails to increase the debt limit, it would cause severe hardship to American families,” her statement read.
Both Mr Biden and Mr McCarthy are under pressure from the left and right flanks of their respective parties to hold the line.
With a one-seat Democratic majority in the Senate and Republicans in narrow control of the House, a deal has so far proven elusive.