The former culture minister was sworn in before the ruling family council on Saturday, local media outlets al-Watan and al-Roya reported.
There was no official confirmation of the reports.
The Omani constitution stipulates that a successor must be named by the royal family within three days of the throne falling vacant. The sultan must be a member of the royal family as well as "Muslim, mature, rational and the legitimate son of Omani Muslim parents".
The late Sultan Qaboos, one of the Middle East's longest-serving rulers who maintained the country's neutrality in the region, had been ill for some time and was believed to have been suffering from colon cancer.
The 79-year-old ruled Oman since overthrowing his father in a bloodless coup in 1970.
He had no children and had not publicly appointed a successor to rule the nation's 4.5 million people.
Oman observers had said the sultan's three cousins - including Haitham bin Tariq Al Said - stood the best chance.
Mahjoob Zweiri, a professor at Qatar University's Persian Gulf Studies Centre, said to maintain Oman's status as a "mature state" will be an important task.
"Taking into consideration new challenges including unemployment … the leadership has to take into consideration all those matters," Zweiri told Al Jazeera.
"In regional matters … the role of Oman played so far is significant. It confirmed the importance of Oman regionally and internationally," he said. "I think this is likely to continue".
Zweiri said that Haitham bin Tariq Al Said was "very close" to the decision making process and has been aware of what is involved when it came to mediation efforts in the region.
"I don't expect really any major shift in all of that," he said.
"I think Oman will continue to play that positive role with cooling down a lot of conflicts because they seek stability and security for the region."