Mills moved to The Riverside Stadium on a year's loan in August under Peter Reid's reign.
"We've had discussions with Danny Mills and his agent this week and those discussions are ongoing," Middlesbrough chairman Steve Gibson told the Boro website.
"Leeds have got complex difficulties and time is running out. We would hope to sign him permanently before the end of the week but we might not.
"But Danny is with us until the end of the season. If not now, then we expect a permanent deal to happen at the end of the season.
"We want him to stay, the fans have taken to him and I think he wants to stay."
United want around £1.5m for the England international and this has also alerted Birmingham and Tottenham, both of who are monitoring the situation.
"We've had discussions with Danny Mills and his agent this week and those discussions are ongoing," Middlesbrough chairman Steve Gibson told the Boro website.
"Leeds have got complex difficulties and time is running out. We would hope to sign him permanently before the end of the week but we might not.
"But Danny is with us until the end of the season. If not now, then we expect a permanent deal to happen at the end of the season.
"We want him to stay, the fans have taken to him and I think he wants to stay."
United want around £1.5m for the England international and this has also alerted Birmingham and Tottenham, both of who are monitoring the situation.