Unite general secretary Sharon Graham said healthcare workers do not want to see longer waiting times and cancelled appointments, but told BBC Scotland they may be forced to act.
Unite is balloting NHS members on industrial action after a 5% pay increase was turned down last month.
Ministers said they were disappointed but were “re-engaging” with unions.
Unite said it had amassed a £50 million strike fund and called on the Scottish government to table a “decent” pay offer.
“Strikes are a last resort but obviously I will back my members where they want to take that action,” Ms Graham said.
She insisted that union members “don’t want the consequences” of the pressure on services that strike action would bring.
It comes as Scotland reported its worst ever A&E waiting times last week, and the cost of living crisis is expected to place further strain on hospitals this winter.
She said: “It’s not very often that you get people in the NHS going out on strike. That is the reality, it’s a huge thing for them to do.
“But on the other side of that argument, you’ve got people with families, you’ve got people who can’t feed their families.”
Ms Graham pointed to a Unite survey that suggests more than 152,000 people (24%) in Glasgow are living in food poverty, while more than 82,000 (13%) cannot afford to pay their household bills.
“People feel that they are in food poverty right this moment – they can’t afford to take any more pay cuts.
“Of course we don’t want that sort of disruption, so I would say to the respective governments come to the table and put decent pay offers down.”
Healthcare staff from Unite and four other unions are being balloted on industrial action after rejecting the Scottish government’s pay deal.
Unison, GMB, Royal College of Nursing (RCN) and the Chartered Society of Physiotherapy also condemned the 5% offer.
Health Secretary Humza Yousaf, right, said: “While we respect the mandate given to trade unions, I am disappointed they voted to reject the record 5% pay deal for NHS Scotland Agenda for Change staff, and are now holding ballots for industrial action.
“We are re-engaging with trade unions in the hope of reaching a satisfactory outcome.”
Ms Graham, who spoke to BBC Scotland at Unite’s Scottish policy conference in Glasgow, also took aim at Prime Minister Liz Truss over the UK government’s mini-budget last week
She said the tax cuts for the most wealthy in England and Wales represented a “bonanza for the rich”.
The union leader was also asked her view on a second Scottish independence referendum.
She said: “I don’t think the general secretary in London should be telling people in Scotland what they should or shouldn’t be doing on indyref2.
“In the first instance that’s my Scottish members who will tell me that and from there the Scottish people will decide.”