Andy Powell, 52, and Nazrul Hussain, whose age has not been released, were both confirmed to have died by the bus operator in an internal email sent to staff and seen by the Yorkshire Evening Post.
Andy, a grandfather from Morley who was a passionate Leeds United fan, passed away on November 1. Nazrul, who died on October 19, was a father of five who had already lost his brother to Covid-19.
The Hussains worshipped at Shahjahal Jamia Masid in Harehills. The mosque confirmed his death and that of his brother Mustafa several months previously.
The two separate emails sent to First employees read: "It is with great regret that I must advise of the very sad death of our colleage Nazrul Hussain, driver at Hunslet Park.
"Whilst he had been absent from work with an apparent sickness bug for about a week, the cause of death has now been diagnosed as Covid. He leaves behind a wife and five young children. Our thoughts are of course with his family, who lost Nazrul's brother to Covid some months ago, and of course to his colleagues here at Hunslet.
"It is with great regret that I must advise of the tragic death of another colleague from the Hunslet Park depot, Andy Powell.
"Andy has been absent from work since October 5, having tested positive for Covid-19, and had been in and out of hospital since then, We were, however, hopeful of Andy making a full recovery until recently, but sadly he was readmitted to hospital at the end of last week and died last night.
"He leaves behind a wife and family including a devoted grandson. Our thoughts are with his family and colleagues here at Hunslet."
An employee at the depot claimed that around 50 staff were off sick with confirmed or suspected Covid-19..
First confirmed last week that eight employees based at the Bramley depot had also tested positive.
A spokesman for First said: "We are sad to confirm that two of our drivers have died from Covid-19. Our deepest sympathy and condolences are with their families and friends at this upsetting and difficult time."
"The drivers last worked shifts on Monday October 5 and Tuesday October 13. The NHS Track and Trace service has been notified.
"The safety and welfare of colleagues and passengers remains our utmost priority.
"First West Yorkshire has implemented a range of measures at its depots to help protect the safety of staff and customers. This includes enhanced daily cleaning of offices and engineering departments using a long-lasting anti-viricidal sanitiser, social distancing and wearing of PPE in designated work areas.
"Buses undergo the same daily cleaning regime in the depot and at service breaks and interchanges during the day, which includes disinfecting all touchpoints throughout the vehicle. Plastic screens separate drivers from passengers and customers are encouraged to pay by contactless, where possible. Drivers also have access to hand sanitiser and gloves and use a face covering when not in the cab area."