According to reports in Italy, the Red Devils are keen to beat rivals Liverpool and Arsenal to the Brazilian's signature.
The 31-year-old's deal at Barca is set to expire at the end of the season and will be available on a free.
The proposed £150,000-a-week deal would match the contract of record transfer signing Angel Di Maria, who joined last summer for £59.7m.
GETTY
Manchester United are keen to beat Arsenal and Liverpool to Barcelona's Dani Alves
We'd like to stay, but we'll see
Dinorah Santana
Only Robin van Persie, Wayne Rooney and Monaco loanee Radamel Falcao earn more at Old Trafford.
However, Alves' agent Dinorah Santana insists her client is keen to remain at the La Liga giants.
She said: "He will give his all while he is here, that's all I can say. We'd like to stay, but we'll see. We have until May to renew but if he needs to move, he'll move."
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