Premier League side aim to outbid Celtic and Rangers for £1.75m striker
The Old Firm have been heavily linked with the 20-year-old Stoke City ace.
Campbell is out of contract at the end of the campaign, which would allow Celtic and Rangers to sign the player for just £400,000 due to cross-border compensation rules which benefits Scottish sides.
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Hide AdSheffield United have now entered the race for the player, who has been at Stoke since joining from Manchester City for a reported £1.75million back in 2016.
Campbell has netted five Championship goals this term, four arriving in his last five.
The Blades hope to do a deal this month with Chris Wilder a fan.
The Sheffield United boss revealed the club are hoping to add more players before the transfer window closes.
He said: “I believe we will get one or two in definitely in the next week. As we all know it is a moving market.”