The defender, 29, who has made 36 appearances for the Owls this season and has been a regular starter, is out of contract in the summer.
"I'd definitely like to stay here longer," he told the club's official website.
"Hopefully I'm going to stay and we'll have a more successful season next season.
"I feel like I've done well this season, but the important thing is how the team are doing and the aim now has to be to stay up and prepare for the next season as a Championship club."
Wednesday take on derby rivals Barnsley in a crunch relegation clash at Hillsborough on Saturday and Buxton believes the result could have a huge bearing on his side's survival chances.
"It's going to be a huge game and I'm sure fans will be there in their numbers," he added.
"We've beaten them at their place already this season and we need to beat them again.
"We're going to have to scrap and fight for everything and hopefully we can get what we need.
"We have to play the teams down there with us, starting with Barnsley on Saturday, so it's in our hands what will happen.
"If we beat those teams then we will have earned the right to stay up."