The Black Cats have made an enquiry to Fulham over the availability of the highly-rated forward.
Dempsey is expected to leave Fulham after having been left out of Fulham’s first three games of the season.
Liverpool have been strongly linked with a move for the United States international in recent weeks.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.