The England international has impressed since arriving in Serie A on a year-long loan deal over the summer.
Smalling has slotted smoothly into Fonseca’s team, putting in a series of assured defensive performances before scoring his first goal for the club in Wednesday’s 4-0 win away to Udinese.
Fonseca has asked Roma sporting director Gianluca Petrachi to accelerate his efforts to agree a permanent deal with United.
Smalling’s agent held a meeting with Petrachi last Friday at the club’s training ground.
And on Wednesday, the club made its first contact with the Red Devils by putting a €10 million bid on the table.
Negotiations are likely to drag on, as United are demanding €18m for the centre-back.
The outlet believes that there will be further developments as soon as next week, with the likelihood being that the two clubs agree on a fee of around €15m.
Smalling is said to have made it clear that he would like to remain in the Italian capital, which could help reduce his parent club’s demands.
Petrachi is also in the process of reaching an agreement with the defender’s agent over the salary he will receive once he has been signed.
The former Fulham man has been an ever-present under Fonseca since reaching full fitness at the end of September.
He has made seven appearances in all competitions, scoring one goal.
Smalling was left stunned last week when Scottish referee Willie Collum awarded Borussia Monchengladbach a last-minute penalty in their Europa League encounter at the Stadio Olimpico.
Collum penalised the centre-back for handball, but replays showed the ball had struck him in the face and made no contact with his hand or arm.
With no VAR to overturn the call, Smalling and Roma were left powerless as they watched Lars Stindl fire in a late equaliser from the spot to draw the game 1-1.
FOOTBALL MANAGER 2020 12 teams we can’t wait to be
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