The Kingstone Press Championship side said in a statement the club will "very shortly be unable to fulfil its financial obligations".
It said it had sought insolvency advice as there was a "very real prospect" the club could "cease to exist".
It said the situation was a result of several factors, including the loss of a significant investor.
Mark Aston, the club's head coach, said: "There is still potential here.
"It breaks my heart, it's not easy at the moment."
The Eagles, a part-time side since 2000, went full-time this season in a bid to win promotion to the Super League but missed out on a place in the qualifiers.
A club spokesman said: "The club was totally committed to achieving the goal of playing Super League football once again this year, but due to a complex set of reasons including the strong competition in the league we have seen our best efforts fall short.
"As it stands today, the club is working incredibly hard to ensure it can honour all of its financial commitments in order to see the current season through and also to attract investment and funding from the business or sporting community.
"We will do everything in our power to make sure that this wonderful club lives to fight another season and secures its long-term future."
He said the club was in "advanced discussions" with prospective investors and was talking with Sheffield City Council about the club moving in to the city's Olympic Legacy Park.
The side, one of the founder Super League clubs, has been playing its home fixtures "on the road" since the closure of Sheffield's Don Valley Stadium in 2013.
Rugby league club Sheffield Eagles has said it is in a "dire financial predicament" and could close.