The post office expects to lose $6 billion this year and is seeking help from Congress to fix its finances.
Barnett said the increase, if approved, would generate $2 billion annually in revenue for his agency.
The agency last raised postage rates on Jan. 27, including a penny increase in the cost of first-class mail to 46 cents.
Congress is considering cost-cutting moves that include ending Saturday mail delivery and most door-to-door delivery. The agency says ending Saturday mail delivery would save $2 billion each year. But many lawmakers, along with postal worker unions, have resisted such changes, saying they would inconvenience customers.
The Postal Service supports the proposed delivery changes. It also is seeking to reduce its $5.6 billion annual payment for future retiree health benefits. It missed two of those $5.6 billion payments last year, one deferred from the previous year, and is expected to miss another at the end of this month when its fiscal year ends.
Postmaster General Patrick Donahoe was to appear before a Senate panel on Thursday to press lawmakers for swift action on legislation to fix his agency's finances.
Donahue has said that without help from Congress, the agency expects its multibillion-dollar annual losses to worsen. He has warned that the agency's cash liquidity remains dangerously low.
The Postal Service is an independent agency that receives no tax dollars for its day-to-day operations but is subject to congressional control.