Glazer, who owns NFL franchise Tampa Bay Buccaneers, is already the second-largest shareholder in United with a stake of 19.17 percent.
The family has held talks with US and European banks about possible financing for a take-over, the newspapers said.
Negotiations with the Wall Street banks are being led by Glazer's 36-year-old son, Joel, though so far financiers have been wary of the opaque structure of the Glazer family finances, the paper said.
Among European banks, Commerzbank, which has a long-standing relationship with the Glazer family, is said to have shown the greatest interest so far.
There has also been speculation about a possible bid from Irish racing tycoons JP McManus and John Magnier, who are the club's biggest shareholders with a stake of 28.9 percent stake through their company Cubic Expression.