JENKINTOWN, Pa.—Hillary Clinton has yet to secure the delegates to clinch the Democratic presidential nomination, but the front-runner is increasingly turning to the themes likely to dominate a potential-general election campaign against Donald Trump.

In a campaign swing across Pennsylvania on Friday, Mrs. Clinton trained her fire not on her Democratic opponent, Vermont Sen. Bernie Sanders, who is trailing in the delegate count and facing an uphill battle, but on Mr. Trump, the Republican front-runner who remains her most...