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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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?