The Telegraph says United sent scouts to Norway on Wednesday evening, to watch 15-year-old Odegaard make his international debut.
The youngster became his country's youngest-ever player, winning his first cap at the age of just 15 years and 253 days.
He could now figure against England next month, having spent time on trial at United last December.
"I think he came out of it well and he was very close to deciding the game," said Norway boss Per-Mathias Hogmo. "He showed the kind of qualities he has."
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