The Sun on Sunday reports (June 10th, page 60) Sunderland are 'stepping up a bid' for Naismith, who played under the Mackems' new boss Jack Ross at St Mirren.
And it is claimed Ross is trying to move out players ahead of a £200,000 move for the 23-year-old.
Naismith impressed despite Ross County's relegation last season and has also attracted interest from Aberdeen, Burton Albion and Plymouth, according to the Daily Record.
And the Scottish newspaper suggests Sunderland may also face competition from two lower-end Championship sides to sign the former Scotland Under-21 international.
Sunderland released their long-serving right-back Billy Jones last week, but still have Donald Love and Adam Matthews on the books in his position.
Naismith played under Ross for two months at St Mirren, before being prised away by his current employers.
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