Vice President Biden is weighing a potential 2016 presidential bid as Democratic front-runner Hillary Rodham Clinton stumbles. (Pablo Martinez Monsivais/AP)

As Vice President Biden weighs a possible run for president, personal issues stand as the biggest unresolved obstacle, with Biden trying to gauge whether his family is emotionally prepared for a grueling campaign while still grieving the recent death of his son Beau, according to people familiar with his deliberations.

Biden is now leaning more toward running than he was earlier in the summer, though he is still weeks from a decision. He thinks his White House experience over the past 6  1/2 years, coupled with his grounding in middle-class issues during a long career in the Senate, makes him well equipped to serve as President Obama’s successor.

But following the loss of Beau Biden in May, the elder Biden is concerned about whether his relatives could handle a bid for the presidency and its time demands on the family patriarch. Advisers know that only the vice president can make the judgment about the readiness of his family.

Biden’s advisers said he is on a timetable to decide about running by the end of the summer. That gives him roughly another month, although a decision could come sooner. Should he run, he would enter as an underdog against former secretary of state Hillary Rodham Clinton, who has amassed a huge campaign war chest and is months ahead of the vice president in organizing in the states with the earliest contests.

Biden would be starting almost from scratch, but the mechanics of financing and staffing a campaign appear less daunting to those around him than they once did. At the same time, the vice president would not have unlimited time to begin assembling an operation.

Clinton has been hurt by declining personal ratings as the FBI looks into the security of a private e-mail server she used as secretary of state. This has helped fuel outside interest in a possible Biden candidacy. For Biden, one important aspect of his deliberations is assessing just how weakened Clinton is, though it’s doubtful that will be clear before he must make his own decision.

One factor that is shaping the conversations about a possible Biden campaign is that so much of the conventional wisdom about 2016 already has been upended, whether by the support for the candidacy of businessman Donald Trump or by the challenge from Sen. Bernie Sanders (I-Vt.) to what many said was the invincibility of the Clinton machine. The unpredictable political climate is both a spur and a warning as Biden tries to decide.

As speculation about a possible Biden candidacy intensified, the vice president met with Obama for one of their regular weekly lunches Monday, the first time they have been in face-to-face contact since the president left for his vacation two weeks ago. The session offered Biden an opportunity to update Obama on his thinking about a possible third presidential campaign, and there was no indication that the president did anything to discourage Biden from continuing to deliberate.

White House press secretary Josh Earnest issued strong praise for Biden on the president’s behalf, saying again that Obama believes his selection of Biden was “the smartest decision that he had ever made in politics.”

Obama has remained neutral in the contest for the Democratic nomination, though he has consistently praised Clinton for her work as secretary of state and as a prospective president. Earnest declined to be drawn into a discussion of the president’s leanings in a hypothetical Biden-Clinton contest. He would not rule out an eventual presidential endorsement of one of the candidates.

The press secretary’s comments about Biden were not significantly different from what the president has said in the past. But in the current context, Obama’s language, and that of his spokesman, are being closely watched by those partial to Biden or Clinton.

Biden would face financial and organizational hurdles if he launched a campaign at this point. (Jonathan Ernst/Reuters)

CNN reported that Biden and his team met Monday night with former Obama White House advisers Anita Dunn and Bob Bauer at his residence.

Talk about Biden’s intentions ramped up over the weekend after it was disclosed that he met Saturday with Sen. Elizabeth Warren (Mass.), a favorite of the progressive wing of the Democratic Party.

Biden, who asked for the meeting, wanted to talk to Warren about middle-class and working-class issues, which have become her trademark in the Senate. Biden and Warren are not close personally, but one link between them is Ted Kaufman, one of Biden’s closest advisers, who replaced him in the Senate when Biden became vice president. As a member of the Senate Banking Committee, Kaufman helped draft what became the Dodd-Frank law revamping financial regulations, while Warren served on the congressionally chartered panel overseeing the implementation of the 2008 Wall Street bailout; Kaufman replaced Warren on that panel when his Senate term ended in late 2010.

Advisers to Warren played down the significance of the meeting, noting that she did a similar session with Clinton months ago.

Many progressives had hoped that Warren would run for the Democratic nomination, but she has demurred. In her absence, Sanders has appropriated much of the energy on the party’s left, which has boosted his support in the polls. What would happen to that support, should Biden run, is unknowable, according to the assessment of those familiar with Biden’s deliberations.

The vice president and a small group of advisers have been surveying the political landscape over the past weeks, trying to assess the organizational and financial hurdles he would face in mounting a late-entry campaign against Clinton, and discussing the rationale for a possible candidacy.

The team has done some outreach, but much of the communication has been in the opposite direction, as fundraisers and others have offered encouragement and a willingness to help with a possible Biden campaign.

Though Clinton has a huge head start, there is confidence that plenty of talent is still available for a Biden campaign team. Among the issues up for discussion is how difficult it would be to put together an operation that could compete effectively against Clinton in the Iowa caucuses, where organization is considered one key to success.

Biden’s advisers seem less worried about New Hampshire, because that state holds a primary rather than caucuses. The vice president also is optimistic about his prospects in the South Carolina primary, the third contest on the 2016 calendar.

Clinton’s financial advantage is not underestimated, but a late-starting campaign requires less money than an early-starting one. Clinton raised $47 million during the first three months of her candidacy but also burned through $18 million in that time.

Biden does not have a large network of fundraisers but might be able to draw on a new generation of financial backers. By one estimate, he would need about $30 million for his campaign to get him through the first four contests, but only if he had a substantial super PAC behind him that could raise and spend three times that amount.

The Clinton factor represents one of the trickiest calculations in Biden’s decision-making. Those urging him to run are hardly indifferent to what has happened to her in the past several weeks, and that has clearly helped create more interest in a possible Biden campaign.

But Clinton’s weaknesses are not a rationale for the vice president to run. He would need an agenda of his own, presumably a bold and, loyalists would hope, forward-leaning progressive vision grounded in middle-class economics and his experience in dealing with international issues.

What complicates all this is that neither Biden’s family considerations nor Clinton’s political health will be fully clear by the time the clock runs out and he must declare his intentions.

Paul Kane and Karen Tumulty contributed to this report.