It is understood 245,000 of the union's members voted in favour of taking strike action in relation to the ongoing dispute regarding public sector pensions.
This was a majority of those voting which means workers including council and NHS staff could take part in industrial action on 30 November.
The decision to strike came on account of changes to public sector pensions proposed by the Government earlier this year, but most Unison members are likely to have voted before the new offer was made on Wednesday.
Tom Flanagan, national head of employment law at Irwin Mitchell, said the result of the ballot raises questions and could see a renewal of calls for reforms of strike law.
"It will be interesting to see how Unison moves forward on this. There is no guarantee that strike action will take place. Unions need to give seven days' notice before a strike begins, so Unison has until 22 November to decide how to play it," he said.
"A number of other ballots on this issue are still going on, so Unison may choose to hold back on announcing its plans until the publication of other results of ballots which may be affected by the government's new offer.
"But the turnout in the Unison ballot was only 29% and even though 78% of those who took part in the ballot voted in favour, in effect that is less than a quarter of the voting membership, at only 22.5% This is hardly a ringing endorsement for industrial action. If Unison goes ahead on the strength of this majority, there may be renewed calls for reforms to be introduced similar to those flagged over the last twelve months by the CBI and in the Report of the think tank Policy Exchange.
"These could include requiring a minimum percentage of the workforce or union membership to vote in a ballot or to vote in favour. For instance, at least half the workforce are able to vote (in other words are union members) and/or the turnout should be at least 40% of those entitled to vote. That latter requirement would mean that the Unison ballot result would not constitute sufficient support for strike action.
"There is undoubtedly an argument that organisations - and potentially the whole country - should not be brought to a halt on the vote of a small percentage of the workforce."
