The 29-year-old has been out of football for more than a month when a two-month stay with Derby County ended on transfer deadline day in March.
Blades boss Neil Warnock has confirmed that Whelan would undergo a medical before having a trial with the club, with a view to next season.
He told the club's official website: "Noel knows one or two of our lads and he is trying to get a contract for next season. We've told him that he can do pre-season with us and it gives us both a chance to have a look at each other, but it's a casual arrangement."
Whelan started his career with Leeds before moving to Coventry City in 1995, before moving onto Middlesbrough in 2000.
He has scored 50 league goals in just over 250 appearances.