Downing Street described last year’s election of Nicolas Maduro, which has plunged the oil-rich country into economic freefall, as “neither free nor fair”.
And the prime minister’s spokesman said: “We fully support the democratically-elected national assembly, with Juan Guaido as its president.”
Emmanuel Macron, the French president, did similar, saying he saluted the courage of the “hundreds of thousands” of Venezuelans, who were marching for freedom.
However, No 10 stopped short of the US president’s move of recognising Mr Guaidó as the president, saying: “The UK position is to recognise states rather than governments.”
Mr Maduro was sworn in for a second term this month after elections last May that were marred by an opposition boycott and widespread claims of vote-rigging.
But hyperinflation, power cuts and shortages of basic items have driven millions of people from the country.
At a rally on Wednesday, Mr Guaidó, who is the head of the opposition-controlled National Assembly, told the cheering crowd that protests would continue “until Venezuela is liberated”.
Mr Guaidó then raised his right hand and said: “I swear to formally assume the national executive powers as acting president," vowing to lead a transitional government and hold free elections.”
In an apparently coordinated statement, Mr Trump described Mr Maduro’s leadership as “illegitimate”, adding: “The people of Venezuela have courageously spoken out against Maduro and his regime and demanded freedom and the rule of law.”
He said he was not considering military action, but added that “all options are on the table” – and called on other nations to follow suit in supporting Mr Guaidó.
Asked about the crisis, the prime minister’s spokesman said: “The 2018 presidential election in Venezuela was neither free nor fair, so the regime’s basis for power is deeply flawed.
“We fully support the democratically-elected national assembly with Juan Guaidó as its president.”
The spokesman also criticised Caracas for expelling US diplomats, saying: “In relation to the United States, we think it’s totally unacceptable for Venezuela to cut off diplomatic ties.
“The solution to this crisis lies in working to find a peaceful and diplomatic solution not in expulsions.”
And, strongly backing Mr Guaidó, he added: “The Venezuela government needs to respect the authority of the national assembly.”
Later on Thursday, the foreign secretary Jeremy Hunt issued a statement echoing the comments from Downing Street, and said the UK does not regard Mr Maduro as the "legitimate leader" of Venezuela.
During a visit to Washington for talks with US vice president Mike Pence, Mr Hunt added: "We are extremely concerned about the situation in Venezuela, but it is clear that Nicholas Maduro is not the legitimate leader of Venezuela.
"The election on May 20 was deeply flawed; ballot boxes were stuffed, there were counting irregularities and the opposition was banned.
"This regime has done untold damage to the people of Venezuela, 10% of the population have left Venezuela such is the misery they are suffering.
"So the United Kingdom believes Juan Guaido is the right person to take Venezuela forward. We are supporting the US, Canada, Brazil and Argentina to make that happen."
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