Obama and Biden, who served as his vice president, sat down facing each other in chairs from across a room for a "socially distanced" conversation.
Both men released the video on Twitter, where Obama has 120.8 million followers, the most on the platform.
"Can you imagine standing up when you were president, saying, 'It's not my responsibility?'" Biden asked Obama, referring to Trump's efforts to evade blame for the pandemic.
"Those words didn't come out of our mouths when we were in office," Obama replied.
The Trump campaign did not immediately comment.
With traditional campaigning still in limbo due to the pandemic, the video offered a glimpse of how Obama - still overwhelmingly popular among Democratic voters - may be deployed to build enthusiasm for November's presidential election.
During their first in-person meeting since Biden became presumptive Democratic nominee, they spoke about the need to expand on the Affordable Care Act.
They noted that the Trump administration is trying to convince the US Supreme Court to invalidate the ACA, also known as "Obamacare," their administration's signature program that vastly expanded US health insurance coverage.
"It is hard to fathom anybody wanting to take away people's health care in the middle of a major public health crisis," Obama said.
They also discussed Biden's ability to empathize, a trait his campaign has stressed to contrast him with Trump.
"I don't understand his inability to get a sense of what people are going through," Biden said. "He can't relate in any way."Obama said, "It is a sign of leadership when you are willing to hear other people's experiences."