Bamford has been a free agent since the summer after helping Leeds win promotion to the Premier League last season.
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The Blades have endured an awful start to the season, currently sitting in 22nd on just 10 points from 15 games.
Manager Chris Wilder sees an opportunity to lift his attack with the know-how and experience of Bamford, says The Sun.
Bamford was seen mocking Wilder last season after Leeds pipped Sheffield United to promotion. But the striker quickly called Wilder to apologise in the aftermath.
Wilder said at the time: “I had a conversation with Pat yesterday and I’ve no issue at all with it. I’m not sure about the words from a fellow pro, but I appreciate and respected him giving me a call and we had 20 minutes.”