The Pakistani schoolgirl survived a Taliban bullet to her head after being airlifted to Birmingham, where she now lives with her family.
The 18-year-old has already made a visit to the California-based Ivy League university campus, 'The Sunday Times' reported.
Her father, Ziauddin, told the newspaper that a key consideration for Malala had been the weather.
"They have 300 days of sun out of 365 days, which is a rare phenomenon in the UK. Three hundred days of sun makes a difference," he said.
Despite the severe head injuries Malala suffered in the gun attack in her native Swat Valley, northwest Pakistan, in October 2012, Malala has excelled at school in Birmingham where she is studying for A-levels in history, maths, religious studies and geography.
She is looking at university applications after acquiring her A-level qualifications by next year. While Stanford appears to be her favoured choice, Oxford has not been ruled out and she has visited the university with a teacher from her independent girls' school in Edgbaston, a suburb of Birmingham.
Malala, who won the Nobel Peace Prize last year, has voiced an ambition to return to Pakistan one day and become a politician.
Degrees in Oxford's acclaimed PPE (philosophy, politics and economics) would prove ideal for this ambition.
"Once she called me from school and said, 'I am going to play sport. I love my headscarf but playing sport it is a problem to wear'.
I said, 'Jani [sweetheart], it's your scarf. When you feel comfortable, wear it, when you do not feel comfortable, do not wear it'," her father told the newspaper.