Veteran defender Weir is training with United after leaving Rangers last month.
Blades boss Danny Wilson is now considering handing Weir a deal at Bramall Lane after being impressed by the former Everton man in training.
"There is no rush to do anything. He is out of contract and can sign any time," Wilson told the club's official website.
"His travelling is easy and he is very relaxed. He wants to keep playing so he continues to have ambition.
"David has been training with us for the last two weeks, through no commitment, but if anything does happen to anyone in our squad then we need to have our finger on the pulse.
"You look at David and you would think he was 30! He is a terrific athlete and a great model to the younger lads."
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