Staff including cleaners, porters, and catering staff employed by Serco will walk out on 14 and 17 February.
The moves comes after a two-day strike, due to take place last week, was paused at the last minute to allow for talks organised by local MPs Heidi Alexander and Will Stone.
However, Unison says Serco turned up to the meeting ‘ill-prepared and unwilling to make an offer' that could resolve the dispute.
The union says Serco is refusing to pay two one-off bonuses awarded to other NHS staff in 2022/23, worth at least £1,655, which means Great Western Hospital employees have been paid less than their colleagues in neighbouring hospitals for doing exactly the same work.
Unison South West regional organiser Tom Kennedy-Hughes said: 'Serco's refusal to take the talks seriously was an insult to the hardworking staff who keep Great Western Hospital running.
'Serco workers deserve the same pay as their NHS colleagues. It's time the company stopped stalling and made a fair offer.
'It's thanks to Heidi Alexander and Will Stone that Serco came back to the negotiating table. But they need to come back with a serious proposal, or the dispute won't end.'
Swindon North MP Will Stone said: ‘I'm disappointed that Serco were unable to come to the table with a meaningful resolution to this long-running dispute for the hard-working staff at GWH.
‘I will continue to champion their work and press Serco to resolve this dispute and recognise the hard work and risk their workers took through the Covid pandemic.'
In response, a Serco spokesperson, said: ‘We have met with both MP Will Stone and MP Heidi Alexander to discuss how they can support Serco in escalating discussions with the DHSC to fund the one-off payment. They both agreed to assist in facilitating engagement with the department.
‘It is disappointing that further strike dates have been announced and we will continue to work with all parties in trying to find a resolution.'