The Sunday Mirror says United boss Jose Mourinho has urged the club's hierarchy to meet Juventus' asking price of £53m ($73m) for the Brazil international.
The paper adds that the Red Devils are unlikely to move for Sandro until the end of the season but will hasten their approach if Chelsea launch a bid for the defender in the current window.
Juventus are reported to be open to the idea of selling the 26-year-old if they can secure an adequate replacement, with Paris Saint-Germain also monitoring his situation in Turin.
Mourinho has predominantly used Ashley Young at left-back this season, while Luke Shaw has also featured sporadically following his return to fitness from a foot injury.
United have also been linked with moves for Tottenham left-back Danny Rose and Celtic defender Kieran Tierney.
Chelsea and United could go head-to-head to sign Alexis Sanchez, who is in the final six months of his contract at Arsenal and could be acquired for a transfer fee in the region of £35m in January.
However, Blues boss Antonio Conte said last week that he had no intention of pursuing Sanchez, who has also been linked with Premier League leaders Manchester City.
"I have great admiration for great players, I think Alexis Sanchez is one of these players," Conte was quoted as saying by Goal.
"I don't know [where Sanchez will go]. You have to ask him which is his choice, or maybe ask Arsenal because Arsenal has to decide his future before June. For sure, his future will be in a great club."
Chelsea signed Ross Barkley from Everton in a £15m deal earlier this month, with the midfielder expected to make his Blues debut against Norwich in their FA Cup third-round replay at Stamford Bridge on 17 January.