Bamford, 32, is without a club after leaving Leeds by mutual consent over the summer.
The experienced forward was in talks with Spanish club Getafe but a move did not materialise and has been training with Championship leaders Coventry City in a bid to maintain fitness.
Chris Wilder wants to sign striker who started viral 'w***er' chant
Blades boss Wilder has bemoaned his lack of a 'number nine' with the club languishing in the Championship's relegation zone.
Wilder left Bramall Lane over the summer following the club's defeat in the play-off final by Sunderland.
However, the veteran coach was back in the dugout after replacement Ruben Selles' lost all six of his games in charge.
Leeds and Sheffield United have shared a local rivalry over the years, which in recent times has intensified due to the two clubs' proximity in a competitive sense.
During the former's promotion celebrations at the end of last season, a video of Bamford went viral appearing to show the striker starting a popular Elland Road terrace chant.
The song in question includes the lyric, 'Wilder is a w***er', which the Sheffield United boss later addressed publicly.
"I'm a big boy, you give it, you take it. You give it out locally, you take it locally as I have always done. We have our moments, other teams have their moments."
Patrick Bamford starting off the chantDU DU DU WILDER IS A WANKER pic.twitter.com/n8Jspp8n6PApril 22, 2025
Wilder also revealed Bamford had been in touch, explaining his actions: "I'm not sure about the words used from a fellow pro, but I appreciate and respected him giving me a call."
Now, it appears Wilder would like to add Bamford to his Blades squad, which suggests there is no ill feeling between the two.
Sheffield United are 22nd in the Championship after 15 games this season and face fierce, city rivals Sheffield Wednesday following the November international break.
"I would have always been looking for a number nine [this summer]," Wilder said recently. "[We need] a number nine in the building [that] will give us the different options."
"Would I like a No. 9? Yes, to have that on the table to go to. We're chasing games tactically as well, because of the situation that we're in. We're just trying to do something different to try and get back into the games."
Sheffield United's current striking options include Tom Cannon, Danny Ings, Chiedozie Ogbene and Tyrese Campbell.
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