Clinton campaign officials were alert and anxious in late summer 2015 to any sign that Vice President Joseph R. Biden might decide to enter the Democratic presidential primary at the last minute.

In a leaked email, former Clinton White House aide Steve Bachar of Denver sent a warning to John Podesta, who is now campaign chairman for Hillary Clinton.

“Prominent dem consultants in the West are getting calls from [Steve] Richetti asking them to be regional coords/consultants,” Mr. Bachar wrote. Mr. Richetti is Mr. Biden’s chief of staff.

Mr. Podesta replied, “Can I call you tomorrow?”

Mr. Richetti was exploring the possibility of the vice president joining the race as the controversy over Mrs. Clinton’s private email server intensified.

Senior advisers to Mrs. Clinton also fumed about AFL-CIO President Richard Trumka’s praise of Mr. Biden, after the labor chief said on a a Sunday talk show that the vice president has been “fighting for working people his whole life.”

“It implies that HRC has not,” wrote Clinton aide Ann O’Leary in a Sept. 6 email to campaign official Nick Budzinski. “Is there anything we can do to get him to add something nice about HRC? If he says it and then says HRC has too - or gives an example of her fighting for the first responders after 9/11 or something, I think it will help the perception that there is a labor record contrast between the two.”

Mr. Budzinski said Mr. Trumka was “annoying.”

“I’d like to just call the AFLCIO and talk to Mike Podhorzer about this,” he said. “Let me know if anyone disagrees.”

Ron Klain, a Democratic operative who served as chief of staff to Mr. Biden, sent an email to Mr. Podesta showing that his support of Mrs. Clinton’s campaign was having a backlash.

“It’s been a little hard for me to play such a role in the Biden demise — and I am definitely dead to them — but I’m glad to be on Team HRC, and glad that she had a great debate last night,” he wrote to Mr. Podesta and Clinton communications director Jennifer Palmieri.

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