Burnham has agreed to send British military advisors to assist Saudi Arabia, Bloomberg reported, but had not yet committed to sending any additional aid.
The prime minister was also warned by UK officials that the Bab al-Mandeb Strait was critical to global shipping and the economy, and must be prevented from falling into Houthi hands.
The Foreign Office and MoD declined to respond to Bloomberg’s questions.
Earlier on Monday, Bloomberg reported that Saudi Arabia was seeking to increase the amount of oil shipments passing through the Strait of Hormuz, after closing its East-West oil pipeline.