The result of the ballot on the improved offer is expected next Friday (30 September).
300 Menzies' staff, packers, pickers and drivers, belonging to Unite were due to stage 24-hour strikes today (23 September) and Monday 26 September.
Unite national officer for the graphical, paper and media sector Ian Tonks said he couldn't comment on the situation until voting had closed and that details of the improved offer are not being disclosed to the media.
Last April, Menzies staff were granted a new pay offer of £7.25 an hour, just five pence above national living wage, and had their double time and time off in lieu removed for seven of the eight bank holidays.
Since then, they have been balloted twice to strike, the second time being after they were offered a 3.85% wage increase, which Tonks said at the time would be offset by the loss in extra bank holiday pay.
The majority of those who were set to strike are based in either Scotland or the South East. Menzies has its headquarters in Edinburgh.
Its main distribution centres in England are in Ashford, Aylesford, Greenwich, Norwich, Portsmouth and Weybridge. Along with the Edinburgh head office, in Scotland, they are in Dundee, Dunfermline, Glasgow, Inverness, Irvine and Paisley.
In 2015, Menzies turned over just less than £2bn.