When Labour were in opposition, Angela Rayner, addressing the TUC last year, promised to increase SSP - though she did not specify by how much.
It is scheduled to rise by £2 a week from April to £118.75.
The Department of Work and Pensions has highlighted that the Employment Rights Bill will extend eligibility for sick pay, and to make it available from the first day of illness.
Currently, the first three days off work are unpaid.
The unions have welcomed this - but believe the bigger issue is the level of payment.
While many employers have more generous sick pay schemes, around one in four workers have to rely on the legal minimum.
A recent report from Citizens Advice said that SSP was the biggest employment issue their advisers had to deal with.
The Safe Sick Pay campaign co-ordinated the letter to the prime minister.
Amongst the signatories are the leaders of the civil service union, the PCS; the teaching union the NEU; the Bakers' Union, and the postal union, the CWU.
Some of the health service unions have argued that there are wider implications for public health.
Professor Nicola Ranger, chief executive at the Royal College of Nursing, said the government needed to "live up to" its manifesto promise.
"Forcing nursing staff to choose between going to work unwell or struggling to make ends meet if they take leave is not only unfair but a risk to patients too.
"People will only receive £3 an hour when they are off sick."
Chair of the BMA's executive council, Professor Phil Banfield, said: "Moving onto SSP often means a huge drop in income for many people, forcing them to go back to work before they are fit to do so.
"All of this contributes to further physical or mental ill health, and more sick leave."
The Labour-affiliated GMB union didn't sign the letter to the prime minister but it has also expressed concerns.
It conducted a survey of care workers, which suggested one in three of them could not afford to take sick leave. SSP is less than a third of the national minimum wage for over 21s.
The DWP has said it agrees that no one should be forced to choose between their health and financial hardship, and has been consulting on strengthening sick pay.
The TUC response to that consultation stated: "If the government is to fully meet its pledge to 'strengthen' sick pay, the rate of SSP must also increase.
"As it stands, the level is inadequate to meet basic living standards and…. is around 20% of average earnings - amongst the lowest of the UK's European counterparts."