The two leaders discussed Mr Duterte's controversial war on drugs, which has left more than 7,000 people dead in under 12 months.
The North Korea crisis was also raised.
According to the White House, Mr Trump "enjoyed the conversation", and has extended an invitation - although a date has not yet been set.
The visit will be a chance "to discuss the importance of the United States-Philippines alliance, which is now heading in a very positive direction," a statement from the White House said.
A meeting between Mr Duterte and Barack Obama was called off last year after he called the US president a "son of a whore".
Mr Duterte has faced international condemnation for his brutal crackdown on drugs in the Philippines.
Thousands of suspected drug dealers have been killed by police or vigilante groups, and tens of thousands more have been rounded up and arrested.
The president - who, like Mr Trump, was elected last year - has also said he would be happy to kill the country's millions of drug addicts.
Mr Trump is no stranger to controversial comments either, including labelling Mexican immigrants as "rapists" and making derogatory statements about women.
Philippines President Rodrigo Duterte has been invited the visit the White House following a "very friendly" telephone call with Donald Trump.