Besler is in demand on the back of his exploits at the FIFA World Cup, where the United States international defender started all four matches as the country reached the last-16 in Brazil.
The 27-year-old central defender, who has spent his entire professional career playing for hometown club Sporting KC since 2009, confirmed bids had been tabled by Championship outfit Fulham and Premier League side Sunderland.
"I know Fulham has offered Sporting Kansas City. I know that Sunderland put in an offer or is putting in an offer," Besler told Kansas City radio station 810 WHB.
"We're trying to get something done as soon as possible, because it's just a situation that you need to get resolved as quickly as you can, because I want to be staying in Kansas City if I'm going to stay.
"If they're going to sell me to a different team, I want to know that as soon as possible so I can just know that and prepare for that."
Besler - the 2012 MLS Defender of the Year - said he had been in dialogue with USA coach Jurgen Klinsmann for advice about a potential move to Europe.
"Of course, [Klinsmann] wants his players to go over and try and make it and develop and eventually play for an Arsenal or a Manchester United," he said.
"But the thing to realise is he doesn't want a player to just go over to Europe and go to a team at the bottom of the standings, and you're sitting on the bench and you're not happy and you don't like the coach. He doesn't just want that."
