The union has set up a £35m fund for possible strike action after the proposal was made for nursing staff in England.
It has called for a 12.5% pay rise.
The RCN's NI director, Pat Cullen, said the union would await the outcome of a pay review before considering its next steps in NI.
The government has said the 1% offer represents what is "affordable" at a difficult time for public finances because of the coronavirus pandemic.
Ms Cullen, referring to the clap for carers initiative, said "it didn't take very long for government to move the claps to slaps".
She said that members, who "have carried us right through Covid... working 15-hour days on most days", were now feeling angry and let down.
"Did we anticipate this happening as quickly and to be let down as quickly? Probably not, but the government just seems to have a very short-term memory on this occasion," she added.
She said she anticipated the results of this to be known by May or June.
"Let's see what will happen then", she said, referring to the potential for members to be balloted for industrial action.
The prospect of a strike comes just more than a year since NI's largest health unions agreed to end months of industrial action over a pay and staffing levels dispute.
It is almost inconceivable that health workers find themselves in a position where they are talking about strike action again.
In fact, if these threats come to fruition, an already astonishing year will be bookended by walkouts and industrial action.
In the 14 months since, health and social care workers have gone from the picket line to the front line, as they joined the fight against Covid-19.
After the most difficult of years, it does feel scarcely credible that there's friction once again over a pay rise.
The government argues that 1% is all they can offer when there is a public sector pay freeze. But this offer has triggered "dismay and anger" among health unions.
One could argue health workers have the better hand. The support of the public will have been strengthened immensely over the last 14 months and that's something governments across the UK will have to filter into their negotiations.
The pay proposal in England covers most hospital staff, but does not cover GPs or dentists.
A government spokesperson said the plan was a "real-terms increase" as the latest official inflation figure was 0.9%.
RCN general secretary Dame Donna Kinnair said 1% would represent about £3.50 more per week in take-home pay for an experienced nurse.
Stormont's Department of Health said "any decision on a pay award for NI health workers will be taken after the pay review bodies make their recommendations".
"We do not intend to pre-empt those recommendations.
"Our pay award decisions can also only be taken in the context of a wider public sector pay policy set by the Department of Finance."
On Saturday, a joint statement issued by the RCN, Unison, Nipsa and Unite unions said members in NI had responded to the government proposal with "dismay and anger".
"They know their worth and value, as does the public and all those who have been cared for by them over the last year," it said.
Did you hear that we're writing Iraq's new Constitution?
Why not just give them ours? We're not using it anymore.