Washington CNN  — 

Joe Biden’s team appears to be leaning more towards a presidential run than away from one, according to a Democratic source who has been in touch them. This source doesn’t believe a decision has been made but left a conversation with those advising Biden with a strong sense he very well might enter the race.

Biden has been told by aides he should make a decision by October 1st.

A possible plan – if he decides to run – currently involves Biden announcing his intentions in the first week of October, the source said.

Speculation about a Biden run was sent into overdrive when he met Saturday with Sen. Elizabeth Warren, D-Massachusetts, the progressive icon whose decision not to run herself was a major boost for Clinton.

And on Monday, President Barack Obama’s press secretary, Josh Earnest, praised Biden’s “aptitude for the job” and said it’s possible that Obama will endorse a candidate in the Democratic primary.

Clinton, who worked alongside Biden in Obama’s first term as secretary of state, has struggled in recent weeks to overcome the controversy surrounding her use of a personal email address on a private server during her tenure as America’s top diplomat.

RELATED: Biden meets with Warren in Washington

Sen. Bernie Sanders, I-Vermont, currently appears to pose the most potent threat to Clinton’s chances of winning the Democratic nomination, particularly in early voting states like Iowa and New Hampshire. But Clinton’s fundraising and organizational muscle could be tough to surmount over the course of a months-long battle.

After a town hall meeting in New Hampshire on Monday, Sanders said he doesn’t know if Biden will get in but that “if he decides to get into the race, is that I will, as I have done up to now, run an issue-oriented campaign.”

Sanders said, though, “Joe’s views, I suspect, I know on a number of issues, are different than mine.”

And he dismissed Biden’s meeting with Warren, saying, “I have had many meetings with Elizabeth Warren. She is a very good friend of mine. People meet with different people.”

RELATED: Joe Biden hires John Edwards campaign alum Kate Bedingfield

Sanders said he sees Clinton as a vulnerable front-runner but that he doesn’t know whether a Biden run would chip into support for Clinton or himself more.

“We are gaining and I think what the polls seem to indicate is Hillary Clinton’s support seems to be receding a bit,” Sanders said. “But we have a long way to go. Joe would be a formidable opponent. I am not sure who politically it would help.”

Earnest said Obama and Biden were having lunch Monday but that he didn’t know what they’d discussed.

He said it’s what everyone in Democratic politics is “pretty interested to find out – is what decision the vice president is going to make.”

RELATED: Judge orders State Department to seek Clinton server information from FBI

“The President has indicated that the decision he made, I guess, seven years ago, to add Joe Biden to the ticket as his running mate was the smartest decision he’s made,” Earnest said, adding that those comments are a sign of Obama’s view of Biden’s “aptitude for the top job.”

Earnest said it’s not clear whether Obama will weigh in for one of his administration’s members over the other.

“I wouldn’t speculate at this point about whether or not the president would offer an endorsement,” he said. “… I wouldn’t rule out the possibility of an endorsement in the Democratic primary.”

A spokeswoman for Biden said Monday, “The Vice President has not made a decision about his political future. Anyone speculating that he has made a decision is wrong.”

CNN’s David Chalian and Dan Merica contributed to this report.

CLARIFICATION: This story has been updated to more clearly state the potential for a Joe Biden presidential bid.