The Sunderland Echo claimed the new Mackems manager, Jack Ross, was weighing up a move for the centre-back, having also wanted him at previous club St Mirren.
Championship sides Wigan Athletic, Blackburn Rovers and Bolton Wanderers were also credited with interest in Robertson, although it is Rotherham who appear to have secured his signature.
It remains to be seen whether Robertson, whose Blackpool contract expires at the end of this month, was formally approached by any of those clubs, or Sunderland.
The Mackems have already strengthened in the Scot's position, bringing in Alim Ozturk from Boluspor last week.
That said, the Wearside club could still have vacancies to fill there, having seen John O'Shea, Marc Wilson, Tyias Browning, Jake Clarke-Salter and Brendan Galloway depart since the end of last season.