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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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.