The aircraft had to turn back.
"The Saudi authorities refused to give the plane carrying President Bashir permission to cross their airspace," Emad Sayed Ahmed, the presidential press secretary, told AFP on Sunday.
Ahmed said Bashir was not flying in his normal presidential aircraft but was using a plane rented from a Saudi company.
Sudan's leader was travelling to attend President Hassan Rohani's swearing-in before Majlis (parliament) in Tehran.
Rohani later took the oath at a ceremony attended for the first time by foreign dignitaries including regional leaders -- but without Bashir.
Ahmed said that when Bashir's plane entered Saudi airspace the pilot informed authorities that it had approval "and that it was carrying Sudan's leader.
"But they said the plane didn't have permission," forcing it to return to Khartoum, he said.
The official SUNA news agency had sent a brief SMS alert at 0706 GMT announcing that Bashir "leaves for Tehran on an official two-day visit to Iran".
Iranian Foreign Ministry spokesman Seyyed Abbas Araqchi was quoted by the ISNA news agency as calling the Saudi move "very unfortunate", and adding that Tehran was investigating.
HH/HH