Obama told Democratic donors that Bernie Sanders' campaign is running out of steam
Obama also reportedly signaled that Clinton's Democratic primary rival, Vermont Sen. Bernie Sanders, was nearing the end of his campaign.
The Times cited anonymous sources to describe what Obama said, but a White House official also confirmed his comments to the paper.
Obama, who has stayed neutral in the primary, "chose his words carefully, and did not explicitly call on Mr. Sanders to depart the race," The Times' Maggie Haberman and Michael Shear wrote.
"Still, those in attendance said in interviews that they took his comments as a signal to Mr. Sanders that perpetuating his campaign, which is now an uphill climb, could only help the Republicans recapture the White House," they added.
A source told The Times that Obama's tone was urgent as it increasingly appears that the Republican nominee will be Donald Trump.
Clinton, who served under Obama as secretary of state, is facing a potentially extended series of primary contests against Sanders. The senator's campaign has touted its strength in the upcoming primary contests throughout April.
Obama also reportedly indicated that he was aware that some weren't "excited" about a Clinton presidency, but dismissed the perception of Clinton being inauthentic. He cited former President George W. Bush, who at one time was praised for his authenticity.
His comments came days before Clinton's big wins in Tuesday's primaries. She won in all five states, which many observers saw as a clear sign that she will ultimately secure the nomination.