Nyatanga suffered the disappointment of relegation from the Championship last season but is being offered another tilt at the second tier after his contract with Bristol City expired, Sky sources understand.
The Glovers are seeking experience as they prepare for the first Championship campaign in their 118-year history.
That is something that the 24-year-old defender has in abundance, having played his part in Derby County's promotion to the Premier League back in 2007.
Yeovil are also keen on Painter, who is also free agent having spent the last three years at Brighton and Hove Albion, playing over 90 games for the south coast club.
The 26-year-old played a significant part in their promotion to the Championship back in 2011 and Yeovil are keen for the former Birmingham City left-back to boost their bid for second-tier survival.
Painter also played 68 league games for Swansea earlier in his career and spent time on loan at Bournemouth in League One last season.