WASHINGTON—President Barack Obama took a political gamble Wednesday by proposing to curb the growth of Social Security and Medicare, hopeful that the concessions would draw rank-and-file Senate Republicans into a budget deal that has so far proven elusive.

The offer came in the White House's $3.778 trillion spending plan for the year that begins in October, which called for about $1 trillion in tax increases over 10 years and higher spending on programs such as education, transportation and mental-health services.

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