Fells was diagnosed earlier this week with MRSA, a powerful bacteria resistant to penicillin-type treatment. Fells is expected to undergo more operations on his foot, which is at risk for further infection, according to the report.
NFL.com, citing a source with knowledge of Fells' condition, reported early Sunday that doctors hope to save Fells' foot from amputation but also fear that the MRSA has infected his bone and could travel into his blood.
Giants spokesman Pat Hanlon has acknowledged that it "is a serious situation that has been taken seriously from the beginning. We're all fighting for Daniel."
Fells, 32, was placed on season-ending injured reserve Monday and has been hospitalized for over a week.
After Fells' diagnosis, the Giants took precautions to prevent the further spread of MRSA by scrubbing their locker room, training room and meeting rooms, a source told ESPN.
Hanlon also announced that the Giants have consulted with several infectious disease specialists.
"We are working with infectious disease specialists, and we have defined protocols that we are following in consultation with the Duke Infection Control Outreach Network and local infectious disease specialists," Hanlon said Tuesday. "Those protocols are being followed carefully."
Fells posted a message on his Twitter account Sunday, shortly before the Giants' game against the 49ers.
ESPN.com Giants reporter Dan Graziano contributed to this report.