According to several of Sunday's newspapers, United boss Sir Alex Ferguson has asked chief executive David Gill to explore a move for the Argentina striker, who seems certain to leave Upton Park this summer.
According to The Mail on Sunday, Old Trafford officials discussed a deal with connections of Tevez last week and are keen to take the frontman on loan for a season while the complicated issues surrounding his ownership are sorted out.
However, the News of the World claims businessman Kia Joorabchian, who was behind Tevez's transfer from Corinthians to West Ham last summer, has since spoken to officials at Inter Milan.
It is thought the Italian champions would be prepared to pay up to £34million for Tevez's services, eclipsing the £12million that United are willing to pay for a permanent deal.
Real Madrid are also in the hunt and hope to hold talks with Tevez's representatives this week, while Liverpool and Chelsea are also thought to be monitoring the situation.
But Ferguson, who is currently on holiday in the south of France, appears keen to make sure United are involved in the scramble for his signature.
The Red Devils boss has admitted in the past that he was offered Tevez and Javier Mascherano last summer, but turned down the move because of uncertainty over the deal.
United have also been strongly linked with Ajax striker Klaas-Jan Hunterlaar, but appear set to lose out to Juventus in the race for the £10million-rated hitman.
They are also struggling to land other striking targets, including Tottenham's Dimitar Berbatov and Atletico Madrid's Fernando Torres, which may explain why Ferguson has stepped up his attempts to do a deal for Tevez.