Hillary Clinton’s campaign announced a proposal on Wednesday to eliminate tuition at in-state public colleges and universities for families with annual incomes up to $125,000 — largely embracing a core position of Senator Bernie Sanders, who had pledged to make tuition at public institutions free for all students.

While stopping short of Mr. Sanders’s vision, the proposal is likely to hearten many of the senator’s young supporters who had flocked to his insurgent campaign. Mrs. Clinton and her aides are working to unite the party behind her before the Democratic National Convention begins on July 25 in Philadelphia. She and Mr. Sanders have already discussed the importance of featuring the issue of the affordability of higher education prominently in the general election.

Mr. Sanders, who has yet to end his candidacy and endorse Mrs. Clinton, praised her education proposal as a “very bold initiative” in a brief news conference in Washington on Wednesday, calling it a significant step for party unity. “The final product is the work of both campaigns,” he said — a clear sign of thawing relations between the two after a bitterly fought nominating contest.

The policy proposals come as Mrs. Clinton has been engaged in intensely personal sparring with Donald J. Trump, the presumptive Republican nominee, and could be an effort to turn the page in her campaign a day after the F.B.I. director, James B. Comey, was sharply critical of her “extremely careless” use of a private email address and server.