Sky Sports sources understand Celtic have rejected a fourth Norwich bid for Gary Hooper.
Two offers lodged earlier in the transfer window were turned down, and Celtic manager Neil Lennon said this week he would not be putting a price on the club's top scorer.
But that has not stopped Norwich coming back with two further bids on deadline day.
Both have been dismissed, and the Premier League club now have just a few hours to come up with an offer which persuades Celtic to sell.
Norwich have already brought in one striker on deadline day, signing Luciano Becchio from Leeds in a swap deal involving Steve Morison.
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