The exchange was captured on camera when Mr Biden was on his way to meet the Times’ editorial board as part of a series of interviews with Democratic 2020 election candidates.
Ahead of his interview, the former vice president spoke to an elevator operator called Jacquelyn, who expressed her admiration for him.
“I love you. I do. You’re, like, my favourite…” the woman said, before taking a selfie with Mr Biden.
Video of the moment has since been viewed more than 500,000 times in total on Twitter accounts for Mr Biden and Matt Hill, his deputy national press secretary.
The exchange is useful for Mr Biden as he has sought to counter critics who have argued he is the establishment candidate and does not connect with ordinary voters.
On Sunday, the Times endorsed both Minnesota senator Amy Klobuchar and Massachusetts senator Elizabeth Warren for president over Mr Biden, who is currently the frontrunner in polling for the Democratic primary.
The decision to back two candidates from opposing wings of the Democratic Party — Ms Klobuchar is a moderate, while Ms Warren is a progressive — has been described as “utterly confusing”.
In an article explaining its decision, the Times’ editorial board suggested Mr Biden’s current lead in national polls was “a measure of familiarity as much as voter intention” and criticised the framing of his campaign.
The editorial board noted Mr Biden was “a figure of great warmth and empathy” and the candidate with “the greatest fluency on foreign policy” but argued he was not the right candidate for the 2020 election.
“Merely restoring the status quo will not get America where it needs to go as a society. What’s more, Mr Biden is 77. It is time for him to pass the torch to a new generation of political leaders,” the editorial board wrote.
The former vice president has led national polls for almost the entirety of the Democratic primary campaign so far and holds a lead of eight percentage points over his nearest rival, senator Bernie Sanders, according to analysis by RealClearPolitics.
Mr Biden is currently battling to win the influential Iowa caucus on 3 February — the first major contest of the Democratic primary.
Current polling averages in the state put him ahead in a tight race, in which Mr Sanders, Ms Warren and South Bend, Indiana mayor Pete Buttigieg are all in contention.
However, Mr Biden will have Iowa almost completely to himself for campaigning in the coming days as several of his top rivals have been forced to go to Washington to participate in the impeachment trial of Donald Trump.
Mr Sanders, Ms Warren and Ms Klobuchar will all take part in the Senate trial, which will judge whether Mr Trump should be removed from office over charges stemming from his alleged efforts to pressure Ukraine into declaring a public investigation into Mr Biden and his son Hunter.
The president’s alleged efforts are based on discredited claims of corruption against Mr Biden’s son during his time working for a Ukrainian energy company.