With the transfer window set to close at midnight on Friday, the race is now on to land the Scotland striker.
Although he has dropped out of Wolves' starting XI - he was an unused sub in Saturday's 3-1 win over Bristol City - the Midlands club are reluctant to allow the 23-year-old to leave, and have so far resisted Cetic's overtures.
But Molineux boss, Kenny Jackett, wants to revamp his attack to make a push for promotion back to the Championship. And, if the price is right, Griffiths will be allowed to go this week.
But after SportTimes exclusively revealed Celtic were stepping up their efforts to land the player, it has now emerged an initial offer from the Parkhead club had been turned down, with a post on the Wolves Twitter account saying: "Kenny has confirmed that Celtic have made a bid for Leigh Griffiths which has been rejected. Striker remains in squad for Oldham game."
Despite the knockback, however, boss Jackett has said a deal for the Scotland striker to leave Molineux could still be done.
He told the club's Facebook page: "We do understand every player has their price but we are not desperate to sell players here.
"Leigh has gone public to say he is interested in the Celtic situation as most Scottish lads would be when a club of that size come in.
"But the deal would have to be right for us and we will have to see how it goes. If not Leigh is still part of our squad and is with us for Oldham tomorrow and Bradford on Saturday."
Speaking at a press conference today, Celtic boss Neil Lennon revealed that talks between his club and Wolves were "ongoing".
When asked how far apart both parties were, he said: "I have no idea. All I know is there was a bid went in.
"So we have made our intentions clear and the talks will keep going now for however length of time it takes to get a resolution one way or another.
"We will see if we can get a deal thrashed out."
When asked what Griffiths would bring to Celtic, Lennon said: "Goals. He had a fantastic season last year. I think he has improved as a player and there is scope for him to get better."
Aston Villa and Swansea are now both showing interest in acquiring Griffiths, which could push Wolves' asking price beyond Celtic's valuation.
And, after missing out on a foreign goalscorer last week, the man who banged in 28 goals in 42 games while on loan at Hibs last season has jumped to the top of their list. Reports from Israel that the Hoops had been chasing Hapoel Tel Aviv striker Omer Damari and Macabbi Haifa midfielder Eyal Golasa - both in the stable of agent Dudu Dahan - have proved to be inaccurate.
Lennon has also denied any knowledge of reputed interest in Falkirk defender Steven Kingsley and confirmed talks are still ongoing with Georgios Samaras and Joe Ledley over new contracts.
While their futures are unclear, the Hoops boss is confident Fraser Forster and Virgil van Dijk will still be at the club when the window closes, although he is not ruling out bids being made for the star pair.
Despite coming up against brick walls in their pursuit of some targets, the Hoops boss is still hopeful he can get in a striker and a player for another position before the week is out.
Stefan Johansen is the only player bought in this month.
The midfielder, who cost £2million when he moved from Norwegian champions, Stromsgodset, made his debut for the club as a late substitute in their 4-0 thumping of Hibs at Easter Road yesterday.
Lennon promised: "We're breaking him in gently, and there will be a lot more to come from Stefan."
The result propelled Celtic 15 points clear in the title race, and well on course to retain the championship before the split.
However, Lennon has warned his Bhoys they will have to maintain their own high standards if they are to avoid disappointment in their quest for an unbeaten league campaign.
He acknowledged his side had been under quite a bit of pressure before Van Dijk struck a wonder free-kick to put them 2-0 up with 14 minutes to go, and said: "A lot of that was down to Hibs, but I think we encouraged it by being sloppy.
"We looked a little bit ragged at times with our play. So, when we changed it for the last 20 minutes, we looked a lot better again.
"It is obviously about decision-making at times, and it is something that gives you food for thought as we go on."