Story highlights Wasserman Schultz has been criticized for her leadership of the DNC

Sen. Dick Durbin was floated as a replacement

Washington (CNN) Rep. Debbie Wasserman Schultz appears safe as chairwoman of the Democratic National Committee until the November elections after Democrats privately mulled options to replace her, including with Senate Minority Whip Dick Durbin, according to several sources familiar with the discussions.

Wasserman Schultz had come under withering criticism from Sen. Bernie Sanders and his supporters who accused the Florida congresswoman of being unfairly biased toward Hillary Clinton. Sanders, who publicly backed the congresswoman's primary foe , demanded new leadership at the DNC as he began negotiations with Clinton over his endorsement of his presidential bid.

But those talks didn't pick up much steam, sources said, with Clinton showing little appetite to boot Wasserman Schultz and cause a distraction, while Sanders focused his energy instead on influencing the party platform.

In the aftermath of the rowdy Nevada Democratic convention in May, where angry Sanders supporters heckled Clinton surrogates, Democrats on Capitol Hill grew increasingly nervous that the same episode could play out at the Philadelphia convention this month. And they privately worried that Wasserman Schultz would not be able to control the outrage of Sanders' supporters who felt party leaders tilted the scales toward Clinton.

So privately, Senate Minority Leader Harry Reid offered an alternative: Durbin as Wasserman Schultz's replacement, according to several people familiar with the situation. In an interview on Tuesday, Durbin confirmed to CNN that he spoke with Reid about the matter, but downplayed the discussion.

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