The Easter Road club rejected an initial £2million bid from Parkhead on Friday afternoon following an opening round of negotations.
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It is believed those talks could be kick-started again today and the Glasgow giants may yet land their man at around £2.5m, with issues such as subsequent payments or taking a fringe player on loan, coming under discussion.
However, West Brom are also keen, with Mowbray having worked with the player during his spell in Leith. Celtic remain favourites to land the 21-year-old if Hibs are persuaded to sell and the striker may even submit an official transfer request this morning.
Fletcher has already made it clear he wants to s i g n f o r Gordon Strachan.
'Steven does have an ambition to move to England at some stage but the prospect of Champions League football and a massive club like Celtic is very persuasive,' said a source close to Fletcher. It is understood West Brom's offer is an initial loan deal with a fee built-in for a transfer in the summer. But the Midlands club are in danger of dropping out of the English Premier League after winning promotion under Mowbray, which might put off Fletcher himself.
Hibs insist they want £4million for the player, who made his Scotland debut against Croatia in March last year in George Burley's first match in charge.
And Strachan remained silent on the issue yesterday after the 0-0 draw with Inverness. When asked about Fletcher, he said: 'This is not the right time to talk about that.' Gary Caldwell knows what it is like to make the short journey along the M8 from Easter Road to Celtic Park.
And the central defender has no doubts over Fletcher's ability to cope with a move from Hibs to Celtic - or the striker's desire to move up the footballing ladder. Fletcher went public over the weekend, insisting that the time was now right for him to move on, and Caldwell - who left Easter Road forr Celtic Park i n the summer of 2006 - is full of sympathy for the forward.
'You can outlast a club,' said Caldwell. 'Steven is maybe at the stage where he's done that, where he needs the challenge that Celtic would bring.
Hibernian's Steven Fletcher battles with Celtic's Scott Brown
'If he feels it's his time, I'm sure he will be pushing to get away. I definitely understand it.
'I'm sure he could cope with playing for us. It's a big challenge, you realise that as soon as you walk in to Celtic. If he comes, he w i l l have to deal with that pressure. I'm sure he's ready for it. 'He is very talented and scores a lot of goals. If he comes, he would be a good addition. We'll see what happens in the next 24 hours.'
Celtic could have done with a hungry and skilful forward to apply some finishing touches to their play yesterday, Scotland regular Caldwell admitting that the SPL champions had fallen short of their usual standards.
Yet he argues that an Old Firm win in their next league outing will right all wrongs - and take Celtic that bit closer to a fourth consecutive title.
'We were sloppy and we didn't deserve to win the game,' he said. 'Our quality of passing wasn't good enough and, if we don't pass well enough, we won't win games.
'Going to extra-time on Wednesday night shouldn't be an excuse. We're all fit enough.
'This is a team game and we all drop points together. If people underperform, we have to raise our game. There is no hiding place.
'We're still in front of Rangers and have to come back figthing.
'We've won three straight titles and we want to win it again.
'We have to take responsibility as players. The manager sets us up, gives us everything we need. Once we cross that line, we have to be better than that.
'You have to look at your own job and get that right. We'll do that, each and every one of us.
'It's Rangers next in the league. If we win that, everything will be great again. There is no panic. 'The experience we went through last year wil help us. I don't know what it is but, the tighter it gets, the better we seem to get.
'In the pressure games last year, when we had to win, we seemed to win then. But we have to be better.
'Hopefully, it starts next week in the Scottish Cup - and then carries on against Rangers.' Celtic will be hoping to add to t h e i r squad b e f o r e t o d a y ' s deadline and Wigan midfielder Jason Koumas was also linked with a loan move to Parkhead yesterday.
But Caldwell said: 'That's up to the manager. He has signed big players at the right time in the past. If he does it this time, great. If he doesn't, we have the players who have won titles before.'