Mr Borrell did not reveal any details of the attack, but an EU spokeswoman said the ambassador was "OK".
"Security of diplomatic premises and staff is a primary responsibility of Sudanese authorities," Mr Borrell wrote on Twitter.
Although Mr Borrell didn't name him, the EU's ambassador to Sudan is Aidan O'Hara, an Irish diplomat.
The BBC has contacted Mr O'Hara's office.
Mr O'Hara trained as a lawyer in Dublin, before he started his career in Ireland's foreign office in 1986.
Before moving to Sudan, he worked as EU ambassador in Djibouti and Irish ambassador to Ethiopia and South Sudan.
EU spokeswoman Nabila Massrali told AFP "the security of staff is our priority" and that the EU delegation has not been evacuated from Khartoum following the attack.
She said they were assessing their security measures.
Mr Borrell said the attack on Mr O'Hara was a "gross violation of the Vienna Convention", a United Nations (UN) agreement that outlines the protection of diplomats working in other countries.
Monday was Sudan's third day of fighting between the army and a paramilitary group called the Rapid Support Forces (RSF).
Both sides claimed to control key sites in Khartoum, where residents sheltered from explosions.
The conflict has seen numerous air strikes, artillery fire and heavy gunfire in civilian neighbourhoods.
The US State Department spokesman, John Kirby, said there were currently no plans to evacuate US personnel, despite the because of ongoing security concerns and the closure of Khartoum's airport.
But he urged all Americans to treat the situation "with the utmost seriousness".
The conflict has forced many civilians to shelter in their homes amid fears of a prolonged conflict that could land the country in deeper chaos.