The 30-year-old is a free agent after a short-term deal with Barnsley came to an end earlier this summer.
Turner, who moved from Stirling Albion to Everton in 2003, had endured a 22-month injury nightmare before joining Barnsley. A loan move to Dunfermline from Preston was cut short in March 2012 due to a back problem.
Sheffield United manager Nigel Clough, who is also running the rule over former Derby goalkeeper Ross Atkins, said: "We're taking a look at Iain and Ross and then we'll be making a decision pretty soon."
Turner has one Scotland B cap and was included in full international squads during Craig Levein's stewardship.
The former Motherwell striker, Michael Higdon, joined the club last week