Their calls follow reports that the state-owned bank wants to shed up to 30,000 posts over the new few years.
The Unite union said it planned to meet Royal Bank of Scotland (RBS) next week as it warned of the effect uncertainty had on workers and their families.
Scottish Labour's finance spokesman Iain Gray also called on the bank to detail how many workers in Scotland could be affected and what kind of support it intended to offer them. RBS has struggled since it had to be bailed out by the taxpayer for £46 billion during the banking crisis in 2008.
Since then the banking giant, still more than 80% owned by the taxpayer, has got rid of around 40,000 of its staff.
It has also undertaken a massive restructuring pro-gramme after the Chancellor George Osborne announced its days as a global investment banking powerhouse were over.
Last month, it admitted it expects to announce pre-tax losses of between £7bn-£8bn for 2013. It surprised the City after it revealed a string of scandal-related financial charges had cost the bank more than £3bn.
They include an agreement in December to pay US regulators $100 million (£62m) following allegations RBS broke sanctions against a host of countries, including Libya and Iran.
Other recent problems include an IT failure that left customers unable to use debit or credit cards.
Alongside confirmation of the losses, an announcement on some of the bank's riskier activities is expected when it posts its end-of-year results next week.
It is not thought specific job cuts will be set out at that stage.
RBS is understood to employ around 120,000 staff, meaning a loss of 30,000 jobs would cut its headcount by around one-quarter.
Heavy job cuts are expected in RBS's investment bank business, which it is thought will pull out of its US and Asian markets businesses. It is expected about 18,000 jobs will be lost with the sale of the company's US retail bank Citizens.
Further reductions will also come from its Williams & Glyn's businesses, which employ about 4500 staff.
In a video posted on the company's website, the chief executive of RBS, Ross McEwan, said: "My aspiration is not to run the world's biggest bank. My aspiration is to run the best bank in the UK - nothing to do with size.
"A lot of our costs are old costs related to a big global group that we are not any more."
Unite said it planned to meet RBS bosses next week to demand clarity over UK staffing levels. National officer Rob MacGregor said: "Our message to RBS is that if it wants to deliver on that commitment then it needs all its staff and they need to be treated with respect, rather than being used as disposable pawns."
Mr Gray said staff deserved more. "If this speculation is correct then we need to know where the job losses will come from, how many workers in Scotland will be affected and what support packages will be put in place by the company to help those facing redundancy," he said.
A spokesman for RBS last night said the bank would not comment on speculation.
l Royal Bank of Scotland banking group customers will be able to use Post Office branches to place deposits into their accounts from later this year.
Personal and business customers of RBS as well as its sister banks NatWest and Ulster Bank will be able to pay in cheques or cash in any of the Post Office's 11,500 branches.