A revised order has been expected from the White House since the earlier ban was blocked by a federal court.
The previous order suspended the entire US refugee resettlement programme and blocked citizens of seven Muslim-majority nations from entering the US.
It sparked confusion at airports, as people with valid visas were turned away, and mass protests.
President Trump’s administration argued that the ban was necessary to keep the US safe from terrorism.
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What is different about the new order?
While the text of the new executive order has not yet been published, Ms Conway, a senior aide to Mr Trump, told Fox News that Iraq would be left off the list of countries whose citizens will banned from the US.
Citizens of Iran, Libya, Syria, Somalia, Sudan and Yemen, the other six countries on the original list, would once more be subject to a 90-day travel ban.
However, according to a fact sheet for Congress seen by the Associated Press, the travel ban will not apply to those who already have valid visas.
Iraqi immigrant Hameed Darwish addresses the media after being released from Terminal 4 at John F. Kennedy International Airport during the Donald Trump travel ban in Queens, New York (January 28, 2017)Image copyrightREUTERS
Iraqis, such as former US army translator Hameed Darwish, will no longer be subject to a travel ban, according to reports
Ms Conway also said that green card holders (legal permanent residents of the US) from the named countries will not be affected by the new order.
Refugees would be banned from entry for 120 days, according to Reuters. But Ms Conway indicated that Syrian refugees would no longer be indefinitely barred.
Why were those seven countries singled out?
When will the new ban take effect?
Khaled Haj Khalaf kisses his granddaughter Shams after she arrived with her mother and father at O’Hare Airport on a flight from Istanbul, Turkey on February 7, 2017 in Chicago, Illinois.Image copyrightGETTY IMAGES
This Syrian family were reunited at Chicago’s O’Hare airport in February
Ms Conway said that the order would take effect on 16 March.
The 10 days’ advance notice may help to avoid some of the chaotic scenes at US airports that occurred on 27 January when the first executive order was announced without warning.
Travellers with valid visas who were in the air at the time of the order found themselves detained by border officials on arrival.
At the time, Mr Trump defended the lack of notice, tweeting that “if the ban were announced with a one week notice, the ‘bad’ would rush into our country during that week”.
Media captionMeet the Arab American family divided over Trump
Will the new executive order face legal challenges?
Possibly. The delayed implementation will help the legal case, as one concern of the federal judges who refused to reinstate the original ban was that the justice department had failed to show that the executive order gave enough “notice and a hearing prior to restricting an individual’s ability to travel”.
Judges also found “no evidence that any alien from any of the countries named in the order” had committed a terrorist attack in the US, so the new ban may be subject to similar scrutiny