Mr Stiglitz said the UK should not use its “scarce resources” chasing a post-Brexit agreement.
President Donald Trump has previously said he was working on a “major trade deal” with the UK.
But Mr Stiglitz said the UK should not trust the President over trade talks.
Speaking to Today on BBC Radio 4, Mr Stiglitz, who is also a former World Bank chief economist, said he thought Mr Trump would “talk about making a deal”.
He questioned whether anything Mr Trump negotiated would have “much support in Congress”
In July, International Trade Secretary Liam Fox went to Washington to discuss prospects.
He was pushing for a deal that would boost trade between the UK and the US and take effect when the UK leaves the European Union.
However, President Trump has made it clear he wants less generous trade relations with other nations and trading blocs.
He wants to renegotiate the North American Free Trade Agreement (Nafta), a deal that dramatically reduced barriers to commerce between the US, Mexico and also Canada.
Some experts fear that the Trump administration might be prepared to bypass World Trade Organisation (WTO) rules and impose new barriers to imports regardless of the organisation’s rules.
Asked on Today if the UK should trust President Trump when it comes to negotiating a trade deal, Mr Stiglitz said: “No, I think it would be a waste of time to begin negotiations with him.
“If you look at what he has been putting forward as a negotiating position in the Nafta discussions you get a feeling of how much of a waste of a time it would be.
“He’s proposed that Nafta be a five-year agreement.
“Businesses can’t move forward with the confidence that there’s going to be another agreement in another five years, it is so absurd even the business community who supported him so much have come down and said this is a non-starter.”
Mr Stiglitz also said: “What Trump has done is thrown a hand grenade into international trade relationships, said… he’s going to renegotiate even WTO and he’s put a block in the reappointment of the judges in the WTO.
“Given the hurdles that the UK has in negotiating with the EU… I think it would be just a waste of time for you to use your scarce resources to try to have a trade deal with the United States,” he added.