The Norwegian is desperate to return to the Premier League following the Blades' relegation last season.
The Gunners interest in Berge seemed to have cooled in recent weeks, but their pursuit of Blades goalkeeper Aaron Ramsdale has brought him back into the conversation.
Mikel Arteta's side have no intention of meeting United's valuation of £40m for Ramsdale.
The Sheffield Star says the north London club could attempt to strike a deal which would involve both Berge and Ramsdale.
Officials at the Emirates believes the prospect of a massive lump sum for a Championship club might lure the Yorkshire outfit to the negotiating table.
While Blades boss Slavisa Jokanovic has stated both players are happy at Bramall Lane, there is no doubt they would welcome a move back to the top-flight.
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