Taskmaster is one of Marvel's very unique and very cool villains, but one with whom casual fans are unfamiliar. Director Joe Carnahan aimed to change that a while back, working with Fox to develop a Taskmaster film in after originally being lined up to work with Marvel for a limited-run Taskmaster comic series. In a 2010 interview with SuperHeroHype, Carahan revealed a bit of his vision for the character's film:
"For Alex (Young, producer of "The A-Team") I was going to do Taskmaster. Originally I wanted to do Juggernaut 'cause I had a really good take on that, and then Taskmaster was something we talked about, and I haven't discussed it with him for ages, but I would love to get back in. I love the idea of a guy with photographic reflexes, he can see something and repeat it, I thought that was such a cool idea. I think there's a Moon Knight series where he just beats the sh*t out of Taskmaster and kind of ruined him, and it's a cool character. I think it might be one of those Marvel characters you really need to reimagine because he's got the whole skull and cape-it's a bit grandiose I think-but it would certainly be a cool thing."
Alex Young was the co-president of production at 20th Century Fox until being moved into a role as a producer where he worked on films such as A Good Day to Die Hard and Unstoppable. All signs would indicate that Taskmaster's rights rest with Fox. Rumors swirled that Taskmaster would show up on Agents of S.H.I.E.L.D., but those have since been proven false. Given the likelihood that Marvel Studios do not own his rights, it wouldn't have been possible anyway. It seems very strange that Taskmaster, a villain who was introduced in the Avengers comics, would show up anywhere but the MCU, but there's no real clarity as to why the rights were at Fox.
Outside of the interview with Carnahan, a quick search did not supply any other information about the rights to Taskmaster reverting back to Marvel.