Department of Justice officials have filed legal notices announcing their intention to challenge the block.
The 6 March order placed a 90-day ban on people from six mainly Muslim nations and a longer ban on refugees.
But judges in Maryland and Hawaii questioned the legality of the ban, which critics say is discriminatory.
Their blocking rulings earlier this week were warmly welcomed by civil liberties groups and rights campaigners.
They argued that the temporary ban on people from six predominantly Muslim countries - Iran, Libya, Syria, Somalia, Sudan and Yemen - was a violation of the First Amendment, which guarantees religious freedom.
President Trump insists the move is to stop terrorists from entering the US and has complained of "unprecedented judicial overreach".
White House Press Secretary Sean Spicer said the government would "vigorously defend" the president's latest executive order.
The US government has said it will appeal against a ruling by a judge in Maryland that has blocked President Donald Trump's latest travel ban.