The Blades are in the process of securing the long-term futures of several of their players, with captain Billy Sharp and Martin Cranie the latest to get new deals.
Now Blades boss Chris Wilder says talks with wing-back Stevens, 26, and “a couple” of his Blades team-mates are ongoing.
The Republic of Ireland international - who previously had a loan spell with Doncaster Rovers - has impressed since joining United from Portsmouth in 2017.
"Talks are still ongoing with Enda and a couple of others, too, that I'll keep under wraps,” said Wilder.
“Hopefully in a week or two we can announce those.
"I've always said that we're looking to tie the players down and not let their contracts run down, and it's an ongoing process.
"There's been a lot of talk recently about Arsenal and Aaron Ramsey and his contract running down... every club has their own way of going about it, but we're trying to build and part of that is keeping our best players, and rewarding them.
“Being nice and stable in terms of our process, going forward."
Wilder revealed his second-placed Blades have no injury problems ahead of their trip to Swansea City on Saturday.