The meeting, which will continue into Saturday, is the first Group of Seven (G7) outing for Steven Mnuchin, Trump's Treasury Secretary and a former Goldman Sachs banker tasked with delivering the US economic agenda.
The doubts over trade has seen the gathering in Bari tipped as being the venue for the latest battle of Bari, a fortified port on Italy's southern Adriatic coast that has seen many power tussles over the centuries.
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But in the run-up, officials from all seven members of the club of wealthy, industrialized nations have indicated that differences over Trump's plans to roll back the boundaries of free trade have been pushed to the sidelines.
Hosts Italy have sought to find areas of agreement between Trump's populist but still vague agenda, and the priorities of Washington's chief European partners (Britain, France, Germany, Italy), Canada and Japan.
"There won't be a specific session on issues of trade and protectionism but it could come up in the broader discussion on the state of the world economy," said an Italian official involved in preparing the meeting.
A senior source from another of the participating countries added: "We are not anticipating being any clearer about what the US administration is planning in terms of trade."
But trade is expected to feature in bilateral talks, particularly with Japan and Germany, both large net exporters suspected by Trump of benefiting unfairly from their respective dollar exchange rates.
Among the issues officially on the table are transnational tax evasion, including the policing of web-based multinationals, combating the financing of terrorism, defending financial institutions from cyber attacks and how to ensure that the benefits of growth are more evenly shared.
The latter is an issue that has been on the G7 agenda since the 2007-08 financial crisis. But it has been given added urgency by Britain's vote to leave the European Union and Trump's election triumph, shocks attributed to the negative impact of recent economic trends on key demographics.
Trump's statements during and after his election campaign suggest he favours "inclusive growth" and Mnuchin, 54, has defined his essential role as delivering higher wages for American workers.
Italy's centre-left government, represented by veteran economist Pier Carlo Padoan, has framed the issue in terms of active steps to reduce inequality, and will be presenting a paper on the topic.
But that may clash with the buccaneering capitalist credo Trump espouses while simultaneously promising to restore prosperity to America's rust belt.
Friction to come?
The meeting is the first since outsider centrist Emmanuel Macron was elected as France's new president, although his country will be represented in Bari by outgoing Socialist finance minister Michel Sapin.
Macron is a free trader but also a strong supporter of European integration who believes the EU should use its global muscle to ensure open markets do not lead to its social model being cannibalized.
Most analysts see his election as strengthening the cohesion of the political leadership of the EU's single market, confounding expectations - which Trump has seemed, at times, to share - of its disintegration.
In a possible sign of friction to come, Mnuchin ruffled the hosts on Thursday evening by bluntly asking Padoan a question about the stability of Italy's banks.
That prompted the Italian minister to mount a robust defence of the sector "intended to dispel erroneous perceptions" of the situation, an Italian official told reporters. "There was a friendly exchange on the subject."