That would mean employers would have to provide appropriate training for staff to deal with mental ill-health.
The companies behind the letter argue the promised change in the law would help break the stigma of mental illness at work.
More than 50 leaders of some of the UK's biggest employers have signed a letter urging the government to make mental health first aid mandatory for all workplaces.
Its signatories called on May to follow through with a pledge from the Conservatives' 2017 General Election manifesto to "amend health and safety regulations so that employers provide appropriate first aid training and needs assessment for mental health, as they now do for risks to physical health".
"By investing in the physical and mental health of our people we will not only unlock human potential in the workplace but reduce astronomical costs to the economy", Keenan said.
"Success will ensure every employee has the right to a mentally healthy environment".
The letter argues that the cost should not be an issue to employers, as workplace mental health issues cost the United Kingdom economy nearly £35bn a year, with 15.4m working days lost to work-related stress, depression or anxiety.
It says around 300,000 people lose their job each year due to a mental health problem.
While Theresa May has said she will introduce new legislation to protect against this for schools and employers there has been little progress on the latter.
'This is just one part of improving approaches to workplace mental health, but it represents an important step forward, ' she said.
May said past year she would shake up mental health service provision, which she described as "one of the burning injustices in our country". "It is imperative the government takes notice of United Kingdom employers and the public, who are standing up and calling for change".
Fionuala Bonnar, chief operating officer for Mental Health First Aid England, said the open letter showed business leaders "clearly recognise the need to support their employees' mental health in the same way they do their physical health".
Stephen Clarke, chief executive of WH Smith, added: "We are calling for this legislative change, alongside many other leading employers, as we firmly believe that everyone should have access to first aid support for their mental health regardless of where they work".
Earlier this year, a study by the charity Mind revealed that nearly half of United Kingdom workers have experienced a mental health problem while at their current job. Fewer than half of those had opened up to their employer, suggesting that as many as one in four workers were struggling in silence with issues such as anxiety, stress or depression.
"That's exactly why we're taking forward all 40 recommendations of the independent Stevenson Farmer Review of mental health and employers".
At the last general election, the Conservatives said they would amend health and safety rules so employers would have to treat mental health the same way they treat physical health.