The Brazilian centre-back, 20, is considered to be one of the best young players in his position on the planet, and recently impressed as he helped his side to knock Chelsea out of the Champions League.
United chief Louis van Gaal wanted him over the winter, but a deal failed to come to fruition and Marquinhos has since extended his stay at the Parc des Princes by committing to a new contract.
However, this has not put off the Premier League giants, who remain keen on a summer deal.
Reports claim that £40.1m has been set aside for the Brazil international, with massive wages set to be offered.
LVG is believed to be chasing new defenders ahead of the looming transfer window and is ready to build on the £150m spent last year.