United are reportedly interested in Fielding, 33 - currently at Stoke City - as cover amid doubts regarding the future of Blades custodian Robin Olsen, whose loan from Roma could be cut short.
Aston Villa are keen on signing the Swedish international, according to reports.
Fielding is on a short-term deal at Stoke, with Derby County having also been linked with a possible move.
Meanwhile, Browne has been told by Boro chief Chris Wilder that he can leave the club to boost his prospects of regular football, with Wilder revealing that there is interest from one or two clubs in the wingman.
On the situation with Olsen and the links with Fielding, Heckingbottom said: That’s all it is, links.
"We have to make sure as a club we’re in a position where we can bring someone in. I spoke with Robin, he understands and everyone understands. That’s as far as we’ve got.
"Frank’s on a short term at another club, we’ve been in contact with the goalkeepers we think are available and until we know we can bring one of them in, we can’t let Robin go. I understand his and Roma’s position and we need to respect that. He’s under our loan contract."
And on a potential move from Browne, Heckingbottom - whose side visit Preston on Tuesday night - added: "I keep getting names chucked at me and I tend to only hear them in press conferences..."