It is understood that the 44-year-old has agreed terms with the Championship side and they hope to conclude a deal in the next 24 hours.
Stubbs led Hibs to their first Scottish Cup final success in 114 years earlier this month.
The Scottish Championship side "reluctantly" gave the Englishman permission to speak to the Millers on Monday.
He will replace Neil Warnock after the veteran manager left at the end of the season having kept the South Yorkshire side in the second tier.
Stubbs took over at Easter Road in June 2014 following the side's relegation from the Premiership.
They finished second and third in his two seasons in charge but were unable to gain promotion via the play-offs on both occasions.
Hibs chief executive Leann Dempster said in a statement on Monday that they remained hopeful Stubbs would decide to stay in Scotland.
"I believe that our best chance to keep Alan focused and positive about his job at Hibernian is to allow a brief window within which Alan can speak to Rotherham," she said.
"Reluctantly, therefore, we have agreed to allow that to happen and we will take stock after that.
"While we have agreed to the request, we hope that Alan will see a brighter future here at Hibernian with the continued full support of the board."
If Stubbs does leave he will forego the chance to take his side into next season's Europa League.