The Daily Star says the Black Cats are eager to reduce their high wage bill and are keen to offload the former Middlesbrough man.
According to reports, Everton tried to lure Johnson to Goodison Park on loan earlier this week but their approach was rejected by the club as they are unwilling to subsidise his wages.
Reds boss Brendan Rodgers is desperate to strengthen his attacking options as his side bid to improve on their second place finish last season.
Rodgers could offer Italian striker Fabio Borini to the Black Cats as a makeweight in the deal.
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