Neil Lennon has not given up hope the 23-year-old Scotland striker could become a Bhoy by tomorrow.
But he confirmed there is a valuation for Griffiths which the Hoops are not prepared to go beyond.
The League One club have rejected a second offer - understood to be £500,000.
Now Lennon has decided to set the wheels in motion in the pursuit of another name on their wanted list.
When asked if the club had made overtures for other players, Lennon replied: "Yes, we have. We did it yesterday."
Lennon would not name his new target, but did confirm he was a foreign striker and that he would like to get an answer on him as quickly as possible as the transfer window moves into its final two days.
He said: "We started the ball rolling on him yesterday.
"It is a very delicate situation at the moment, and I need to see where we are on it."
Although Lennon will be hugely disappointed if the window closes without anyone coming in - he accepts Celtic will not be forced to pay what they consider to be over the odds for Griffiths, or any other player on their list.
Wolves have been playing hard ball, even though they yesterday bought striker Leone Clarke to augment their attack.
And, while Lennon has not shut the door on Griffiths, he insisted he can't leave all of his eggs in one basket.
The fact Celtic have turned their attention elsewhere could attract Wolves' attention, but Lennon admitted: " I don't know where we are with the negotiations for Leigh, if they have started up again or not.
"Peter didn't intimate that to me yesterday.
"So, whether we decide to have another go today or tomorrow remains to be seen.
"But, we do have other options for other players."
He continued: "The situation with Leigh is that we have made a bid to Wolves because we like the player.
"It has been rejected and we are debating wether we go in again."
Lennon had also hoped to bring in defensive cover during this window, but is now becoming resigned to the fact this looks unlikely.
He said: "There is nothing jumping out at us at the minute, so I would say that ship might have sailed.
"The defence is playing well, and they are consistent.
"Adam Matthews isn't that far off returning now after his injury."
Lennon confirmed that talks with Joe Ledley - a scorer in the 4-0 rout of Kilmarnock last night - and Georgios Samaras over new contracts are continuing and will do so even after the window closes tomorrow.