"Don't destroy the Democratic Party to satisfy the secretary's ambitions to become president of the United States," Sanders campaign manager Jeff Weaver told CNN.
"We want to have a party at the end of this that we can unify," he added.
After Sanders's win in the Wisconsin primary Tuesday, the campaigns are shifting to New York's April 19 primary, where Clinton is looking to keep Sanders from gaining any more ground via the Empire State's 247 delegates.
Sanders's campaign has vowed to fight to the convention, while Clinton's team has pointed to delegate math showing the Vermont senator has a narrow path to surpass her.
Weaver said Sanders and Clinton should generally engage in a "tough debate" on issues, but he added, "Let's not denigrate other's supporters and tear the party apart."
"The senator has tried to run an issue-oriented campaign; obviously the contrast has become sharper of late," he said on CNN. "We're fully prepared to engage in that environment if they want to."
On Tuesday night, as he was being declared the victor in Wisconsin, Sanders faced intense criticism on gun control from the daughter of a woman who died in the 2012 Newtown, Conn., school shooting.
"Shame on you," Erica Smegielski tweeted at Sanders, linking to an image of the front page of the New York Daily News ripping Sanders for his remarks regarding victims of gun violence having the ability to sue gun manufacturers.