Silverwood's side head the Division One table by 41 points with five games remaining, the first of them at home to Somerset starting on Monday.
Only Notts, in 2005, have won the title following promotion the previous year.
"In our first year back up in Division One, to bring it home would be pretty special." Silverwood told BBC Essex.
Essex lifted the trophy six times in the days of a single-division Championship, most recently in 1992 when Graham Gooch was captain.
Now, current skipper Ryan ten Doeschate has the task to leading the team to their first silverware of any kind for seven years following the end of their involvement in this summer's T20 Blast.
Only two of Essex's last five matches are at home, and they will have to manage without England batsmen Alastair Cook and Tom Westley, their two leading run-scorers in the Championship, because of the current Test series against West Indies.
"We are in a great position, but there's still a lot of work to be done and we won't take anything for granted," said Silverwood.
"We have got five games of Championship cricket left and we can look at them with a lot of excitement. We'll go out there and give every ounce of energy, blood sweat and tears.
"It's the holy grail, and certainly in our dressing room and among the members, although it's great to win (other) competitions, we want to bring that one home."
Essex head coach Chris Silverwood has set his sights on "the holy grail" of winning the County Championship before their return to red-ball action.