The Netherlands international was initially reluctant to join the Premier League giants but the presence of his former Ajax manager Erik ten Hag, who was appointed last month, was a major factor in his change of stance.
United made an opening offer of €60m plus €10m in surcharges on Friday and although this was immediately rejected, talks between the two clubs are set to resume next week.
The Portugal international was heavily linked with a move to Barcelona last summer as he wanted to play in La Liga.
Convinced by Pep Guardiola to stay a year, Silva played an influential role in his side’s fourth Premier League triumph in five seasons.
The former Monaco midfielder is currently on international duty with Portugal and revealed he will try to finalize his future in the summer.
He said: “Unfortunately, I can’t say anything. I’m concentrating on the national team. When the season ends we’ll see what happens.’
Guardiola, meanwhile, has already addressed fresh speculation linking the 27-year-old with Barcelona, saying: “(One move), it will be very difficult.”
Barcelona have also been linked with Wolves midfielder and Silva’s international team-mate Ruben Neves while they remain hopeful of securing a deal for Robert Lewandowski after missing out on Erling Haaland, who joined City.
MORE : Barcelona reject Manchester United’s opening bid for Frenkie de Jong and set asking price
MORE: Frenkie de Jong says yes to Manchester United move to Erik ten Hag at Old Trafford
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