'We may be pushed back to the picket lines', RCN Wales Director Helen Whyley told ITV Wales
The union, which represents the vast majority of nurses in Wales, said its members voted "overwhelmingly to reject" the offer.
But the Welsh Government - which has been negotiating on a joint basis with health unions - has confirmed that the offer has been narrowly accepted by the majority of unions.
The unions form the Wales Partnership Forum which also includes the Chartered Society of Physiotherapists, the British Medical Association and the Royal College of Midwives (RCM) among others.
The RCM said in its two-week online consultation, which followed a pause in planned strike action earlier this month, that 82% of its members rejected the offer.
However, the collective response of the trade unions means discussions will now take place over the implementation of the offer, as well as over pay for 2023/24.
The Welsh Government had put forward an additional pay offer of 3% following a number of walkouts, including by other healthcare professionals such as ambulance workers.Of the 3% pay offer, 1.5% was consolidated, meaning it would have been permanently added to salaries, while the other 1.5% was non-consolidated, meaning it would have been given as a one-off payment.
RCN Wales has asked the Health Minister to return to negotiations within five working days, or future strike action could be on the table.
Helen Whyley, director, said: "The RCN is the voice of nursing - our members make up the vast majority of the nursing workforce in the NHS in Wales.
"They understand best the direct impact it would have, and they have spoken with strength to overwhelmingly reject this offer. We therefore remain in dispute."
Ms Whyley also confirmed she had written to Minister for Health and Social Services, Eluned Morgan MS, asking her to return to the table to negotiate directly with the RCN or risk further strike action.
"If this does not happen within the five working days I set out in my letter, I will have no other option but to announce further dates for strike action," she added.
"Strike action is always the last resort; our members do not want to strike, but this additional offer does not restore the years of being undervalued and understaffed.
"Nursing staff feel, once again, that have left them with no alternative. The Welsh Government must take urgent action to bring this long-standing dispute to a close."
Following a meeting with the Wales Partnership Forum on Tuesday, Eluned Morgan MS said: "Whilst we are pleased that the offer has been accepted, we recognise the strength of feeling amongst members.
"We will continue to work together in partnership to work through the implementation of this offer, and also to begin immediate discussions on next steps including continued meaningful conversations on the nature of the pay award for 23/24 and additional non pay elements to improve the conditions and well-being for our NHS staff."
Responding to the decision by RCN members, Plaid Cymru spokesperson for health and care Rhun ap Iorwerth MS said: "The reality of the additional pay offer was that NHS staff were being asked to accept a further fall in living standards.
"Understandably, they have voted to reject an offer that was little more than a sticking plaster over the deep wound caused by years of underfunding.
"Welsh Government need to get back around the table immediately - that is the only way to avoid further action by staff who are already at the end of their tether."
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