While Mr Biden and Republican leaders may have agreed to a deal at the weekend, they now face the task of convincing hardliners in their party to support the bill.
“Look, you know I never say I’m confident what the Congress is going to do. But I feel very good about it,” Mr Biden told reporters on Monday.
When asked about his message for Democrats who have reservations about the bill, Mr Biden said: “Talk to me”.
House Minority Leader Hakeem Jeffries has not yet publicly stated his support for the bill and Mr Biden is unsure if he can convince progressives in the House, such as Representative Pramila Jayapal's progressive caucus, to back it.
“The answer is: I don’t know. I have a good relationship with Jayapal. I haven’t had a chance to speak to her yet,” he said.
Meanwhile, Mr McCarthy said he expects most Republicans to back the bill. Still, it faces a major test on Tuesday.
Three hardline members in the Republican Party – who have been critical of Mr McCarthy – sit on the Rules Committee responsible for advancing the bill to a full vote on the House floor.
House Freedom Caucus chairman Scott Perry said the debt ceiling deal "totally fails to deliver" on the original positions of Republican lawmakers, pointing to funding for the Internal Revenue Service and by not removing Mr Biden's student loan forgiveness plan from the bill.
"And we will do everything in our power to stop and end it now," Mr Perry said.
Mr Biden and Mr McCarthy finalised their deal on Sunday after reaching a tentative agreement earlier in the weekend.
Should the bill pass, it would suspend the US's $31.4 trillion debt ceiling until January 2025, thus avoiding another showdown until after the 2024 presidential election.
The deal would also boost US defence spending to $886 billion, a 3 per cent increase and in line with Mr Biden's 2024 budget.
Under the deal, veterans' medical care would be fully funded, including funding for the toxic exposure fund from the PACT Act.
The deal also makes it easier for fossil fuel and other energy projects to receive permit approval, one of the key sticking points for Republican leadership.
Unused Covid-19 funding, however, will be clawed back – roughly $30bn of it.