Ross, who has also been linked with Easter Road midfielder John McGinn, was keen to make Kamberi his first signing after quitting St Mirren to take charge of the English League 1 side. But the move for the Grasshoppers forward never got off the ground due to the transfer fee demanded by his Swiss parent club.
The 23-year-old, who still has a year left on his Grasshoppers contract, scored nine times for Hibs after joining them on loan in January.
A transfer fee for the striker to remain in Edinburgh on a permanent basis was agreed at the time the loan deal was made.
Sunderland had hoped to nip in ahead of Hibs, however they were quoted a fee in excess of €2 million and have cooled their interest in the Swiss Under-21 international.
A source close to Kamberi said: “Florian’s form in Scotland had alerted the new manager of Sunderland to his potential and an enquiry was made.
“However, Hibs have a [lower] fee agreed with Grasshoppers, no other club does. It looks like Sunderland’s interest is over.”
Hibs have until 14 June to complete a deal with Grasshoppers for Kamberi on the terms agreed in January.