Sudan wants to be removed from the list so sanctions can be lifted.
The country has been listed since the 1990s when al-Qaeda leader Osama Bin Laden lived there, as a guest of ex-President Omar al Bashir’s government.
Bashir has since been overthrown and relations with the US have eased.
On his part, Mr Pompeo was urging Sudan to normalise relations with Israel.
Mr Hamdok tweeted: “We had direct & transparent conversation regarding delisting Sudan”.
Sudan is named along with North Korea, Iran and Syria.
The country’s leaders are desperate to end the country’s economic isolation and gain access to the dollar-based international financial system to attract loans and investment.
One of the key conditions set by the US for Sudan to be removed from the list was to compensate the families of 17 US sailors who died when their ship, the USS Cole, was bombed by al-Qaeda at a port in Yemen in 2000.
On his stop in Israel, he urged more Arab states to make a peace deal with Israel.
He was referring to the agreement between UAE and Israel, brokered by US President Donald Trump, earlier this month.
We are “very hopeful we will see other Arab nations join in this”, he said.
In a statement on Twitter Mr Pompeo said that he discussed “Sudan’s commitment to deepening the Sudan-Israel relationship” with the prime minister.
But the current transitional government in Sudan “does not have a mandate… to decide on normalisation with Israel”, spokesperson Faisal Saleh is quoted by the AFP news agency as saying.