Blades boss Paul Heckingbottom wants Akpom, 27, and is pushing the club’s moneymen to make an offer — as French side Lens are interested.
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Boro are reluctant to lose Akpom, although there is not much time left on his contract.
If the price is too high United will turn to younger option Archer, 21.
Villa are open to a sell-and-buy-back deal on England Under-21 international Archer — who is also wanted by Middlesbrough, where he starred on loan last season.
Busy Blades are also looking at various targets for midfield reinforcements.
They are in talks with free agent Tom Davies, 25, as a possible addition — but the main snag is the size of his wage demands.
Nottingham Forest’s Lewis O’Brien, 24, is still on their list but they know the figures are too high for them at the moment.
Blades want to keep their salaries in check in case they are relegated.
And Blades are waiting on Manchester City to decide on whether James McAtee, 20, can move on loan or if they could take 21-year-old Tommy Doyle back on the same basis.
BETTING SPECIAL – BEST NO DEPOSIT CASINO OFFERS
New signing Gustavo Hamer was introduced to the Bramall Lane crowd yesterday prior to the defeat against Crystal Palace.
The midfielder arrives from Coventry as a replacement for Sander Berge, who has joined Burnley.
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