US officials had been due in Brussels next week to discuss the deal aimed at boosting bilateral ties.
President Barack Obama earlier cancelled his trip to Asia because of the shutdown.
The US government closed non-essential operations on Tuesday after Congress failed to agree a new budget.
Since then hundreds of thousands of government employees have not been working or paid.
On Friday, US trade representative Michael Froman informed the EU that financial and staffing constraints made it impossible to send a full negotiating team to Brussels.
But he stressed that Washington would continue working with the EU on drawing up the deal, but would have to wait until the shutdown was over.
Reacting to the US announcement, European Trade Commissioner Karel De Gucht said the delay was unfortunate.
“But let me underline that it in no way distracts us from our overall aim of achieving an ambitious trade and investment deal,” he added.
Meanwhile, the White House said that Barack Obama would miss two summits in Asia, including the Asia-Pacific Economic Co-operation (APEC) meeting in Indonesia.
It said the decision was made due to the “difficulty in moving forward with foreign travel in the face of a shutdown”.
The US government partially shut down operations on Tuesday after Republicans who control the House of Representatives refused to approve a budget, saying they would only do so if Barack Obama’s healthcare reform law was delayed or stripped of funding.
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