The dispute has been over changes to "premium payments" for working weekends.
They have been offered a buy-out equivalent to five years' worth of premium payments, and a 4% pay rise.
BBC Wales understands that ministers are providing all of the finance for the extra money being offered.
The previous best offer was a two and a half year buy-out, so Welsh Government is providing another two and a half years' worth of cash.
Staff can take this as a lump sum, or spread over five years. There is an additional option of taking the payment over four years with a pension.
Depending on how many Saturdays and Sundays they work, the payments for affected staff will range from Â£163 to Â£20,000.
The Welsh Government support is significant, as the museum has previously blamed government cuts to its budget for forcing it to cut the wages of some of its lowest paid staff.
NMW in a statement said it had received "additional financial support" from the Welsh Government and has now presented the Public and Commercial Services Union (PCS) with an enhanced offer.
Staff will be required to work no more than one in two weekends as the result of a review of weekend working.
About 300 of the museum's staff receive the payments and 220 are PCS members.
600 staff
300 receive weekend premium payments
Â£4,000 lump sum offer equivalent to two years of weekend payments
-4.7% grant funding cut from Welsh Government, 2016/17
The union has been holding strikes at NMW sites, while industrial action has been going on for more than two years.
NMW said with the pay rise it was a better offer than union members accepted at National Museums Scotland in April.
PCS said: "We welcome the fact the employer has tabled a new offer and we're currently considering the detail and consulting members.
"We remain committed to negotiating a lasting settlement to this dispute that treats our members fairly and with the respect they deserve."
A Welsh Government bail-out has enabled National Museum Wales (NMW) to double its best offer to staff involved in strike action.