In a staff email seen by the BBC, senior managers at Leeds Teaching Hospitals Trust (LTHT) said finances were in 'serious trouble'.
LTHT is one of the biggest trusts in the country with an annual spend of £1.7bn.
Simon Worthington, director of finance at LTHT, said the trust had identified "robust plans" to avoid deficit in March 2025.
Worthington said the trust had identified a need to reduce spending on bank and agency staff after a significant increase in permanent clinical staff.
The BBC estimated 10 other hospital trusts in West, North and South Yorkshire need to save a combined £350m this year.
Director of finance, Gary Boothby, said Calderdale and Huddersfield NHS Trust was targeting savings of £25m in 2024/25, including a reduction in agency and bank staff spend, 'alongside initiatives to transform the way services are delivered in the most efficient way'.
'Providing high-quality care for our patients is our absolute priority, and the potential impact of any proposed efficiency measures will be thoroughly assessed before they are introduced,' Boothby said.
The director said there were no current plans for redundancies with reduction in headcount delivered through staffing vacancies and a 'time-limited vacancy freeze'.