The Dutch defender helped the Rugby Park club secure their place in the top flight next season, but is out of contract this summer.
Allan Johnston, the manager, has offered the 25-year-old defender a new deal, but he seems keen instead to try a new challenge.
"I'm open to everything - a new club would be a nice adventure for me," Tesselaar admitted.
Meanwhile, Kilmarnock's summer signing Jamie Hamill suggested that he would try to convince his former manager at Hearts, Gary Locke, to join him in Ayrshire.
"I would love to see him here," Hamill admitted. "I'll maybe give him a wee phone in the next few days to try and get him involved."