Hibernian say they have “reluctantly” allowed Rotherham United to talk to head coach Alan Stubbs about the vacancy at New York Stadium.
The 44-year-old Englishman led the Edinburgh club to victory over Rangers in the Scottish Cup final.
But his side also failed to win promotion to the top flight for the second season in succession.
“Alan has expressed an interest in hearing what Rotherham have to say,” said chief executive Leeann Dempster.
Rotherham failed in talks designed to retain Neil Warnock at the end of the season after the 67-year-old, who had replaced Neil Redfearn in February, steered them clear of relegation from the Championship in England.
Hibs turned down their initial request to speak to Stubbs at the end of last week.
However, following a meeting between Dempster and head of football operations George Craig, the Scottish Championship club have relented.
“Throughout, we have made it plain to Alan that we want him to remain as head coach of Hibernian and build on what has been achieved,” Dempster said in a statement on Hibs’ website.
“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.
“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.”
Stubbs, who had been on the coaching staff at Everton, took over at Easter Road on a two-year contract following the sacking of Terry Butcher after Hibs were relegated in 2014.
Hibs finished runners-up behind city rivals Hearts in his first season as head coach, but they lost to Rangers in the promotion play-offs.
The Edinburgh side finished third this season under Stubbs, who had signed a contract extension until summer 2017, and again lost in the play-offs, this time to Falkirk.
The season finished on a high note as Hibs went on to beat champions Rangers 3-2 to lift the Scottish Cup for the first time in 114 years.