Silva is set to leave PSG when his contract expires at the end of the season.
Chelsea are desperate for defensive reinforcements after they conceded more goals than any other side in the top 10 of the Premier League this campaign.
To snap up such an experienced veteran on a free transfer is seen by Blues supporters as a bargain.
There are concerns that Silva, who turns 36 next month, may be too old to adjust to the fast pace of the Premier League.
But after impressing in the Champions League finals Chelsea fans have been won over and hope Granovskaia can complete the deal which is already at the advanced stage.
“@ChelseaFC please sign Thiago Silva and make him our new captain. The guy is world class,” one person tweeted.
Another added: “Thiago Silva will be a Good signing and addition to this Chelsea team,
no argument or pundit needed to clarify this.”
A third user wrote: “Thiago Silva is still a class act at 35. He is exactly what Chelsea needs. What a performance by the skipper.”
While a further fan said: “OK I take it back Thiago Silva should come to Chelsea dat guy his actually holding d defence.”
And one more person commented: “I want Thiago Silva at Chelsea. I don’t care about his age.”
Meanwhile, BT sport pundit Steve McManaman also backed Chelsea to complete the deal for Silva regales of his age.
“Football is all about the now,” the ex-Liverpool star said.
“You can’t plan about five, six or seven years because the manager might not even be there then.
“For the here and now he is an excellent defender.”
Chelsea are also hoping to bolster their back line by completing a deal with Leicester for left-back Ben Chiwell