Donald Trump signs the controversial order banning immigration from seven countries
Several other states, including California and New York, are considering whether to launch their own challenges.
But Washington will become the first state to take on the executive order when Mr Ferguson files his lawsuit in in federal court.
Governor Jay Inslee, a Democrat, said: ”It is an insult and a danger to all of the people of the state of Washington, of all faiths.”
He said it was important for the Trump administration to face lawsuits from the state itself and not just cases filed by people who have been impacted by the order.
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It is an insult and a danger to all of the people of the state of Washington, of all faiths
Washington governor Jay Inslee
Tech companies Amazon and Expedia, both of which are based in Washington state's Seattle area, will support the suit and Microsoft said it has been cooperating with Mr Ferguson’s office to provide information about the order's impact "in order to be supportive."
Pressure on Mr Trump has been growing all day with even Barack Obama taking a swipe at his successor.
It is rare for former presidents to criticise their successor's policies so soon after leaving the White House but a spokesman for Mr Obama said he "fundamentally disagrees with the notion of discriminating against individuals because of their faith or religion”.
Barack Obama has taken the unusual step of criticising his successor
The leader of the Democrats in the US Senate, Chuck Schumer, said he would bring legislation seeking to end the ban although it stands little chance of being passed by the Republican-led Congress.
Mr Trump's directive puts a 120-day hold on allowing refugees into the country, an indefinite ban on refugees from Syria and a 90-day ban on citizens from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
The President said the action will protect the US from terror attacks but critics complain it singles out Muslims, violates US law and the Constitution and defiles America's historic reputation as a welcoming place for immigrants.
Tens of thousands of people march against Donald Trump's travel ban policy
The UK Reacts To Trump's 'Muslim' Travel Ban
Mon, January 30, 2017
President Trump signed an executive order banning immigration to the USA from seven 'muslim' countries. This led to protests across America and, now, the UK.
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People demonstrate during a protest at Downing Street in central London against US President Donald Trump's controversial travel ban on refugees and people from seven mainly-Muslim countries
Chaos broke out over the weekend as border and customs officials struggled to put the order into practice amid loud protests at major US airports.
White House spokesman Sean Spicer said Mr Trump's move was popular with Americans.
He said: "The majority of Americans agree with the president. They recognise that the steps that he's taken were to keep the country safe.”