The Sun says Ferguson lost out in his first attempt to sign the 22-year-old last summer, when he was outbid by Real Madrid.
The France striker made a £31million switch from Lyon. But he has been a flop for the Spanish giants, who are now ready to sell for around £25m.
Benzema has started only 14 La Liga games this season and scored just eight league goals.
United's owners, the Glazers, are desperate to make a big-name signing to silence some of the opposition to their reign.