"With regard to comparisons to President Obama's foreign policy decisions, as we've heard before the President fundamentally disagrees with the notion of discriminating against individuals because of their faith or religion," Obama's spokesman Kevin Lewis said in a statement.
This is the first press statement issued by the office of Barack Obama after he left the office 10 days ago.
Lewis said Obama is heartened by the level of engagement taking place in communities around the country.
"Citizens exercising their constitutional right to assemble, organise and have their voices heard by their elected official is exactly what we expect to see when American values are at stake," Lewis said.
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Trump's executive order bars citizens of seven Muslim-majority countries from entering the United States for the next 90 days, suspends the admission of all refugees for 120 days and indefinitely suspends the Syrian refugee programme.
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Meanwhile, defending Trump's immigration policies, White House Press Secretary Sean Spicer said the US President is going to be very proactive when it comes to safety and security of the country and would not wait for things to happen.
"We are gonna put the safety of Americans first. We're not gonna wait and react. The president's gonna be very proactive with protecting this country. We're not gonna wait until we get attacked and figure out how we can make sure it doesn't happen again," Spicer told reporters at his daily news conference.
"How do we get ahead of threats? How do we keep America ahead of the curb when it comes to people who want to do us harm? And that's what the president has done, is he's made sure that every way possible we get down the path of securing this country, putting America's safety and security first and foremost, bar none," Spicer said.
Responding to a question, Spicer said no one knows where the next threat is coming.
"You don't know when the next attack's coming. And so the best you can do is to get ahead of it because if you wait, you're gonna be reacting. And what I think I want to be clear on is the president's not gonna wait," he said.
"I think what I want to be clear about is that since becoming president, he's continued to take steps through executive order and otherwise to make sure that this country is as safe as it can be and that we're ahead of every threat," the presidential spokesman said.