Zuma has been warned by a ministerial task team not to do so.
Sachs resignation follows respected Director-General Lungisa Fuzile's departure in April.
Gigaba said: “Both the Director-General and I are aware of protecting the integrity and transparency of the budget system and process‚ and ensure that all tax and expenditure decision processes continue to be run by the Treasury and Minister of Finance‚ and continues with the consultative process introduced by the first democratic government.
"The Minister wishes to assure the nation that Treasury remains committed to a budget that focuses on fiscal consolidation and mindful of the economic challenges of the country while protecting the poor."
National Treasury said in a statement: "Sachs will not be leaving National Treasury immediately to ensure a proper handover to another senior official and to allow for a smooth transition with as little interruption to the work of the Budget Office as possible."
"Mr Sachs works with many knowledgeable technocrats who he has also helped groom over his years at the department."