Labour and the Liberal Democrats have called for Kemi Badenoch to come clean about her claims of an offer from Stanford University at the age of 16, after former admissions staff said she had described an impossible scenario.

The Labour MP Peter Prinsley has written to the Conservatives leader saying she should lay out the specifics of how the alleged offer came about, given the doubts cast over her story. The Lib Dem education spokesperson, Munira Wilson, said Badenoch risked undermining trust.

Badenoch has defended her claim to have received an offer as a teenager in Nigeria from the elite university to study medicine, sometimes described by her as pre-med, even though the university does not offer that course for undergraduates.

Admissions staff have also said the Conservative leader’s assertion she was offered a place on exam results alone, and was offered a partial scholarship, would not have been possible, with no offers made on that basis.

Jon Reider, the admissions officer during the period Badenoch applied, told the Guardian he had been responsible for international admissions and scholarships and had not offered one to Badenoch.

Badenoch doubled down on Monday, telling reporters she had indeed received offers based on her exam results. “All I will say is that I remember the very day those letters came to me. It was not just from Stanford. I was 16, I had done very well in my SATs,” she said. “But this is 30 years ago, I don’t have the papers, and what the Guardian is doing is reporting on hearsay rather than talking about what the government is doing.”

Multiple former admissions staff and US academics have told the Guardian that Stanford has never made offers based only on SATs – US standardised tests – with no exceptions even for royalty or child prodigies.

Prinsley, a former hospital consultant, wrote to Badenoch to say her claims have been “called into serious question by people in a position to understand the situation, and I would be grateful if you could demonstrate that you have been telling the truth”.

He said the Tory leader should clear up whether she had applied to Stanford and whether a place and financial aid had been offered.

A Labour source said: “Honesty and integrity aren’t optional qualities for those who serve as the leader of His Majesty’s official opposition. The uncertainty surrounding Kemi Badenoch’s Stanford University claims raises important questions that the public deserve to know the answers to.

“Badenoch needs to come clean about what’s happened here and whether she’s been telling the truth to the British people.”

Wilson also wrote to Badenoch saying she should come clean. “If Kemi Badenoch cares about restoring trust, she should start by explaining her own academic record,” she said. “Failing to come clean over these allegations would send a message to the thousands of pupils who just received their exam results that their hard work does not matter and that you can just bluff your way to the top.”