The Black Cats have been pursuing Ayew for several weeks and are understood to have made an initial approach for the player 10 days ago.
The Ghana international has been one of the better performers in an otherwise disappointing campaign for Swansea, scoring eight Premier League goals.
Sunderland manager Sam Allardyce confirmed last week that he still wants to sign a defender and an attacker, having already brought in Bayern Munich defender Jan Kirchhoff and Trabzonspor striker Dame N'Doye this month.
Allardyce had refused to comment publicly on using Borini as part of an offer for Ayew, saying: "I can't speculate on that. Me saying I'm willing to use Fabio would upset Fabio; me saying that Ayew is definitely someone I want for this club means that Swansea could say I have upset this player by saying that I actually want him."
Swansea manager Alan Curtis has previously confirmed his interest in Borini but had spoken of his reluctance to allow Ayew to leave the club as part of the transfer.
"Fabio would be one of a number of players whose name has been mentioned," said Curtis. "But the talk of Andre going there as some sort of part of the deal is wide of the mark.
"I spoke to the chairman just to clarify the situation and he assures me that there is no way that Andre will be allowed to leave during this window.
"On that basis, whether Fabio was part of a deal involving Andre I don't know, but certainly Andre is out of the deal."