Two federal courts shot down the travel ban saying it amounted to religious discrimination, something Dady doesn't disagree with.
The plan, that Mr Trump announced soon after he became President, aimed to stop people from seven mainly Muslim countries entering America. The ban on refugees is also being allowed to take effect on a similar, limited basis.
But it said the ban would not apply to visitors who have a "credible claim of a bona fide relationship with a person or entity in the United States". Saudi citizens are not affected by the travel ban.
The travel ban restricts travel from six Muslim-majority countries in the Middle East and North Africa.
Trump's original order was revised in March to remove Iraq from the list of countries and withdraw the restrictions on green card holders.
In its June 26th ruling, the Supreme Court partially lifted the injunctions issued by the lower courts.
"Then there are stealth attacks through memos being sent by Islamophobes in the white house to different government departments which is being used by some banks and other financial institutions to deny legitimate Muslim charities, the Muslim business, and otherwise traveling Muslims like the son of the great Muhammad Ali", the statement said.
The exception will be those residents of the forbidden countries that have any relationship with the United States citizens, the prohibition against them is declared invalid by the court.
It remained unclear exactly when new instructions would be distributed to embassies and consulates. For entities, the relationship must be documented and formal and must not have been formed for the objective of evading the restrictions of the Executive Order.
Similarly, an applicant's relationship with a distant, nonblood relative in the USA could be considered legitimate.
Last year, about 85,000 refugees were resettled in the country, with more than a third coming from countries affected by the travel ban.
The union that represents Customs and Border Protection agents said Tuesday it wants the administration to provide detailed guidance to help avoid a repeat of the chaos that ensued in January.
But Zarif said the development would force terrorists to do more harm.