Such a pairing would be big news and bring new possibilities to the presidential race and a potential Clinton presidency. But there are also reasons to question whether it would be the kind of dream ticket many progressives envision.
Here are some considerations that might prompt Clinton to pick Warren, and some others why she might just decide to give her a pass.
Warren would not be the first woman picked as a vice presidential nominee -- Democrat Geraldine Ferraro in 1984 and Republican Sarah Palin in 2008 have already secured that honor in each major party.
But a winning Clinton-Warren ticket would make Warren the first female vice president of the United States. After 240 years, the United States would have women filling the top two jobs in the executive branch.
Within her own family, there is a direct precedent for a themed presidential ticket composed of two symbolically similar figures. In 1992, Arkansas Gov. Bill Clinton made a point about his new kind of centrist Democratic leadership by choosing a running mate in his own image: a young, southern politician named Al Gore, then a senator from Tennessee.
Presidential nominees are often motivated in their choice of running mate by finding someone who can paper over their own liabilities with a particular constituency.
Clinton's primary struggle with Vermont Sen. Bernie Sanders clearly highlighted her inability to so far win over many progressive corners of the Democratic Party.
Choosing Warren, a heroine to the progressive movement, could ease that deficit. Warren helped set up President Barack Obama's Consumer Financial Protection Bureau and is a long time anti-Wall Street campaigner who could soothe much of the distrust in the party about Clinton's own ties to big corporate finance.
Warren is also energetic, has no trouble whipping up partisan crowds and has a inside track to big party donors -- all important requirements of a vice presidential candidate.
Vice Presidential nominees are often called upon to unleash the kind of biting political attacks on the rival ticket that would seem unseemly and could compromise the gravitas of a presidential nominee.
Warren already has that role down. She has emerged as the most effective, sarcastic and mocking Democratic critic of Trump's character, record as a businessman and vituperative rhetoric. She is particularly effective on Twitter -- the social media jungle that the real estate billionaire prowls like no other candidate.
In a speech on Thursday, Warren, who the real estate magnate has nicknamed "Pocahontas" because of her claimed native American heritage, castigated Trump as a "thin-skinned, racist bully." And in one recent Tweet, she accused the presumptive Republican nominee of "belching insults" as he could not talk policy.
David Axelrod, a former aide to Obama and senior CNN political commentator said on "New Day" Friday said Warren has worked out a way to distract Trump.
"She has been out there and she has been hitting Trump hard and obviously getting under his skin," David Axelrod, senior CNN political commentator and former aide to Obama, said Friday on CNN's "New Day."
Why Clinton may not pick Warren
If you've made history once -- why bother to do it twice?
By virtue of her gender, Clinton has already ticked the box of having someone on the ticket who can serve as a talisman for women voters as Democrats try to ride a wide gender gap to a third consecutive White House term.
If Clinton sees the vice presidential nod as a way to boost her standing among crucial Democratic constituencies, she might be better served by picking a Hispanic candidate -- perhaps Housing and Urban Development Secretary Julian Castro or an African-American running mate such as New Jersey Sen. Cory Booker.
Vice presidential nominees have sometimes been picked to deliver a crucial swing state or region -- the most famous example is the choice of Sen. Lyndon Johnson to join John F. Kennedy's Democratic ticket in 1960.
If Clinton is struggling to secure Warren's home fiefdom of Massachusetts in the Democratic in November, Trump will already be on his way to being President.
And while Warren is popular with the Democratic left, her strength among Rust-belt white voters who Republicans hope to entice with Trump's earthy appeal, is untested.
Should Clinton play the geographic card with her vice presidential pick, she might be better off opting for someone from a swing state -- such as Ohio Sen. Sherrod Brown, who also has proven blue-collar and progressive credentials.
There's a more fundamental reason why Clinton may discard Warren: the question of whether she is ready for the top job after only four years in the Senate and given her relative lack of exposure to national security policy. She's has also only run one Senate campaign, meaning that throwing her into the heat of a presidential battle, despite her expertize at mocking Trump, may be risky.
While Clinton's pedigree as a former secretary of state and global figure is seen by Democrats as satisfying the "commander-in-chief" test that all potential presidential tickets face, she must also consider the the consequences if she could no longer serve.
"I'm looking at the most qualified people, and that includes women, of course, because I want to be sure that whoever I pick could be President immediately if something were to happen -- that's the most important qualification," Clinton told CNN's Anderson Cooper this week.
The preparedness issue is one reason why some Democrats often mention the name of Tim Kaine as a possible Clinton running mate. The Virginia senator is respected for his foreign policy views. As the commonwealth's former governor, Kaine also has political networks to draw upon in a crucial swing state.
Warren's age may also count against her.
Clinton would be 69 herself when inaugurated president so picking someone from her own generation -- man or woman -- for a sexagenarian ticket might leave the important bloc of younger voters cold.
Though Clinton and Warren met for 66 minutes on Friday in Washington, and have been casual acquaintances, there is not much evidence that they have a close relationship.
The relationship between presidents and vice presidents is difficult to manage -- and it takes time to develop: for example Vice President Joe Biden sometimes infuriated Obama with his lack of discipline during their 2008 campaign. Over the years however, they forged one of the closest bonds of any commander-in-chef and his deputy.
Only Clinton and Warren can judge how their personal chemistry would evolve.