The Blades are also hopeful to draft in a replacement for departed striker Nick Blackman.
Midfielder Robson, 34, has been training with Middlesbrough after leaving Vancouver Whitecaps earlier this month and has agreed to join the Blades until the end of the season.
Leading goalscorer Blackman joined Reading on Wednesday for a reported fee of £1.2million.
"Following Nick's departure we'll certainly be able to bring in one or two players on-loan or short-term contracts," said manager Danny Wilson.
"We're very close to rubber-stamping a move for Barry (Robson), we are just waiting for the international clearance to come through.
"He's a very offensive player, who will score goals and create chances from midfield. He is experienced and a really strong-minded character.
"Barry is comfortable anywhere across the midfield, so his arrival gives us more options than we have at the present time."
Wilson admitted to some frustration during the transfer window as he looks to give his side's promotion hopes a boost.
He added: "Getting players through the door in the timescale we have is very annoying because your targets can soon go elsewhere.
"When a player leaves as late as Nick has in the window it proves very difficult to bring in replacements, but we'll continue to search."