The 27-year-old Sporting Kansas City centerback impressed at this summer's World Cup in Brazil, starting four games for the USA.
Reports have claimed that up to ten clubs from England and Germany have expressed an interest in landing the MLS stopper, but Besler has now revealed that it is Sunderland and Fulham leading the chase.
"I know that Fulham has offered to Kansas City and I know that Sunderland put in an offer or are putting in an offer," he told American radio station Sports Radio 810.
"We're trying to get something done as soon as possible. It's a situation which needs to get resolved. I want to be staying in Kansas City if I'm going to stay. But if they're going to sell me, then I want to know as soon as possible. It's not just my decision. It's the club's decision if they want to sell or not. But I would definitely have an input on a team."