Warren, a leading progressive voice among Democrats, is among those Democratic presidential candidate Clinton is vetting for the vice presidential position, the newspaper reported.
Clinton's rival Bernie Sanders is not, it added. Sources told Reuters earlier this month that Warren, who represents Massachusetts, is considering the potential role.
Representatives for Clinton, Sanders and Warren did not immediately reply to requests for comment on the report.
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Clinton is the Democratic Party's presumptive nominee for the November 8 presidential election, having won the last primary contest this week in the District of Columbia.
Although Clinton and Sanders met this week, the senator from Vermont does not plan to end his campaign or endorse Clinton in a video speech to supporters scheduled for later on Thursday, his spokesperson said.
While the search for a potential partner in the race is still in its early stages, the Journal reported several Democrats said Clinton's campaign is looking at a number of potential candidates, including Warren.
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Other prospective running mates include US Labor Secretary Tom Perez and US Housing and Urban Development Secretary Julián Castro, according to the report.
Senators Tim Kaine of Virginia, Sherrod Brown of Ohio and Cory Booker of New Jersey as well as US Representatives Xavier Becerra of California and Tim Ryan of Ohio are also under consideration, it said. Los Angeles Mayor Eric Garcetti is also a potential candidate, it added.
Warren threw her support endorsed behind Clinton last week as the former secretary of state moved her sights from the nominating contest toward a November 8 match-up against presumptive Republican nominee Donald Trump.
Warren could help Clinton win over Sanders supporters from the party's more liberal wing after a surprisingly protracted primary race.
Sanders, a self-described Democratic socialist, has not yet dropped out. She also would give Clinton a vocal boost in her fight against Trump.
Warren has called Trump a threat to the country and has vowed to keep lashing out at him.
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