The Royal College of Nursing says members are “angry and upset” about the size of the increase.
Nurses may consider industrial action over the offer of a 3% NHS pay rise in England and Wales, a union says.
The Royal College of Nursing (RCN) told the BBC it would be consulting members but staff were “angry” and deserved a bigger increase.
Health Secretary Sajid Javid said the pay rise was a recognition of the “extraordinary efforts” of NHS workers.
Meanwhile, school leaders have said the pay freeze for most teachers in England is a “slap in the face”.
In November the government announced a public sector pay freeze for 2021-22, with exceptions made for those on salaries under £24,000 and NHS staff.
The 3% pay rise is for most NHS staff including nurses, paramedics, consultants, dentists and salaried GPs and is backdated to April 2021.
According to government calculations for the average nurse, this will mean an additional £1,000 a year, while many porters and cleaners will receive around £540.
It comes after heavily criticised proposals made by the Department for Health and Social Care in March said only a rise of 1% was affordable.
The independent NHS pay review body then recommended a 3% rise, which was accepted by the government.
Patricia Marquis, England director of the RCN, said the whole process had been “shambolic” and nurses had “fallen way behind on pay”.
She told BBC Two’s Newsnight: “Our next steps will be to consult with our members about their view on the award.
“And once we have their view – which we suspect will be to say they are unhappy about the level – we will then be considering with them what the next steps might be, which could include consideration of industrial action most certainly.”
She added that nurses were “really angry and upset” at the pay award, which the union says amounts to a pay cut once inflation is taken into account, and it made them feel they were not “valued for what they do”.
The RCN had called for a 12.5% pay increase, which it said would help fill vacancies and retain staff.
Other unions have also criticised the pay rise.
Unison said the increase, though an improvement on the initial offer, fell short of what NHS staff deserved.
The British Medical Association (BMA), which represents doctors, also said the pay rise was disappointing, adding that junior doctors and some GPs could miss out on it altogether.
The GMB union described the increase as “insulting” and said it would recommend its members reject the offer.
Labour’s shadow health minister Justin Madders described the new pay rise as a “U-turn” and called on the government to “make our NHS and key workers feel supported and valued after all they have done for us”.
NHS Trusts and employers said all pay rises need to be fully funded by government, rather than being squeezed from existing hospital budgets.
NHS pay rises are negotiated by independent pay-review bodies that look at evidence from a range of groups before making recommendations to the government.
The pay rise does not include doctors and dentists in training who have their own separate, multi-year contracts.
In theory, the pay-review bodies make recommendations for NHS staff across the UK – but it is up to the individual UK nations to decide whether to accept them.
All NHS staff in Wales will be offered a 3% rise by the Welsh government.
In Scotland, most NHS staff have already been offered a 4% pay rise – backdated to December 2020. This follows a one-off Covid payment for health and social care staff of £500.
Meanwhile, the government has also confirmed that there will be a pause to headline pay rises for the majority of public sector workforces in 2021-22.
But the School Teachers’ Review Body has recommended a pay award of £250 for all teachers earning less than £24,000 in England.
The pay freeze for other teachers was criticised by unions, who said it risked making recruitment challenges worse.
NAHT, the school leaders union, said it was a “slap in the face”, while the Association of School and College Leaders said it was “an absolute insult”, after a year where teachers had worked “flat out” because of the pandemic.
Shadow education secretary Kate Green said: “A real-terms pay cut for the vast majority of teachers is an insult after the heroic work they have done to keep children safe and learning throughout the pandemic.