Democratic presidential front-runner Hillary Clinton Hillary Diane Rodham ClintonJeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Momentum growing among Republicans for Supreme Court vote before Election Day Warning signs flash for Lindsey Graham in South Carolina MORE holds a slim lead over rival Bernie Sanders Bernie SandersNYT editorial board remembers Ginsburg: She 'will forever have two legacies' Two GOP governors urge Republicans to hold off on Supreme Court nominee Sanders knocks McConnell: He's going against Ginsburg's 'dying wishes' MORE in California ahead of Tuesday’s primary, according to a new poll.

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Clinton leads Sanders 49 percent to 47 percent among likely Democratic primary voters in the CBS News Battleground Tracker Poll released Sunday morning.

The majority of Sanders supporters in California, 57 percent, said the main reason they are voting for him is to influence the party, even if they don't think he can get the nomination. Six in 10 Sanders supporters want him to fight on to the Democratic National Convention in late July.

And among California Democrats overall, 71 percent want Sanders to endorse Clinton and have a unified convention when the primaries are over.

Pollsters surveyed 1,187 likely California voters and 674 likely Democratic primary voters from May 31 to June 3. Their findings have a margin of error of 3.9 percentage points for likely voters and 5 points for likely Democratic voters.

Clinton has a wider lead over the Vermont senator in New Jersey, however, at 61 to 34 percent.

CBS surveyed 1,194 likely voters in New Jersey and 586 likely Democratic voters from May 31 to June 3. That poll has a margin of error of 3.8 percentage points for likely voters and 5.4 points for Democratic primary voters.

— Jessie Hellmann contributed to this report, which was updated at 11:09 a.m.