Hunt, who is battling Boris Johnson to become Britain's next leader, raised the murder of Saudi journalist Jamal Khashoggi, the jailing of activists in China and Vietnam and the murder of reporters in Mexico.
Ms Clooney was speaking at the Global Conference for Media Freedom in London, co-hosted by foreign secretary Jeremy Hunt and Canadian foreign minister Chrystia Freeland.
Speaking at the same event, global human rights lawyer Amal Clooney - appointed by Hunt as special envoy for media freedom - went further, suggesting that Trump's rhetoric risked endangering journalists worldwide.
Amal Clooney spekas during the Global Conference for Media Freedom at The Printworks in London, Wednesday, July 10, 2019.
Clooney did not utter Trump's name, but said "The country of James Madison has a leader today who vilifies the media, making honest journalists all over the world more vulnerable to abuse". According to the United Nations cultural body Unesco 99 media workers were killed past year.
She said: "Last year when Jamal Khashoggi, Washington Post columnist, was tortured to death and dismembered by Saudi Arabian officials in Istanbul, world leaders responded with little more than a collective shrug".
The new Global Media Defence Fund - to be administered by UNESCO - will support, train and provide legal support for journalists in the most unsafe parts of the world.
The UK is giving $3.75 million over the next five years while Canada has committed just over $765,050. At its best, a free media both protects society from the abuse of power and helps to release the full potential of a country.
The British government said Sputnik and RT are barred because of "their active role in spreading disinformation".
RT, formerly known as Russia Today, said "it takes a particular brand of hypocrisy to advocate for freedom of press while banning inconvenient voices and slandering alternative media".
The report says the UN Security Council should initiate a follow-up criminal investigation into Khashoggi's murder to build up strong files on each of the alleged perpetrators and identify mechanisms for formal accountability, such as a tribunal.
The United Nations human rights expert who conducted an independent probe into the murder of dissident Saudi journalist Jamal Khashoggi on Tuesday urged the U.S. to act on her damning findings.
Hunt also defended the publication of the private diplomatic cables highly critical of Trump, sent by the United Kingdom ambassador to the US, Kim Darroch.
Ms Clooney warned that Australian Federal Police raids on journalists in Australia and other leading western countries could be used as a blueprint by other nations to clamp down on press freedom.
Asked if he agreed with Mr Trump that journalists were "the enemy of the people", Mr Hunt responded: "I wouldn't use the language President Trump uses and I would not agree with it".