The Sunday Mirror reports that Bruce will step up his interest in the duo this week.
Birmingham winger Larsson is available on a Bosman, while Campbell has a £1million buy-out clause in his contract.
The Black Cats face competition from a host of Premier League clubs for the duo.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.