Blades boss Bryan Robson put tough-tackling midfielder Gravesen, 31, at the top of his wanted list after taking charge at Bramall Lane.
The former Everton star's agent insisted last month that he "wants to stay" at Parkhead and "fight for his place" in Gordon Strachan's plans.
However the arrivals of midfield general Massimo Donati and Scott Brown have seemingly pushed Gravesen further down the Bhoys pecking order.
And Blades chairman Kevin McCabe has reopened talks with the Scottish champions in a bid to tie up a deal.
He told BBC Radio Sheffield: "We've been trying to negotiate terms with Celtic.
"He's on our list, so now it's a question of can we conclude terms and does the player want to come to us?"
Gravesen joined Celtic in a £2million move from Real Madrid last season - but failed to hold down a regular spot.
His agent John Sivebaek is adamant he is keen to be a success north of the border and previously said: "Thomas wants to stay and show everybody is wrong about him.
"He has decided to fight for his place in the team. Thomas loves living in Scotland, he likes the club and fans.
"Celtic have indicated they will accept that. It is hard to imagine Thomas in Sheffield and he has decided to stay."
Vi bruger cookies for at kunne give dig den bedste oplevelse. Ved at bruge vores side accepterer du brugen af cookies. Du kan læse mere i vores persondatapolitik.