Sky Sports says West Brom are in talks with United about a deal for Robinson, as Slaven Bilic looks to strengthen his attacking options in the pursuit of promotion.
The Championship leaders have failed to score more than once in a game in eight of their last 10 matches - a record that has seen the gap to third-placed Fulham cut to just four points.
Robinson fits the profile of what Bilic is looking for - a versatile, wide attacking player who can both score and create goals.
Albion have been long-time admirers - it is thought they were interested in him before he joined the Blades from Preston last summer.
Chris Wilder is still keen to strengthen his attacking options, and a permanent deal is likely if he can sign a replacement, although a loan option is still being considered.