Officials at Bramall Lane inserted a sell-on clause, entitling them to 20 per cent of any future transfer fee, into the deal which took the centre-forward to St Andrews in 2016.
With their Premier League rivals agreeing to sign Adams for a reported £15m, that means Chris Wilder's recruitment budget is set to receive another boost following last season's promotion from the Championship.
Adams, who turns 23 later this month, is understood to have accepted a wage of around £50,000 a week following talks with Ralph Hasenhüttl's side, who visit Bramall Lane in September. He travelled to St Mary's for a medical over the weekend after City sanctioned his departure as part of a summer overhaul designed to help them meet the English Football League's financial fair play rules.
Adams, who scored 15 goals in 55 appearances for United, cost £2m when he joined City.
Meanwhile, United Women will begin the new FA Women's Championship campaign with a trip to Aston Villa. The competition's full fixture list is scheduled to be released later this month.
Kelly Simmons, Director of the Women’s Professional Game, said: "It’s a very exciting time for women’s football with the country currently swept up in the FIFA Women’s World Cup and the Lionesses performing brilliantly.
"We’re looking forward to this increased interest carrying over into our domestic game next season and beyond."