The Co-operative Party has argued that Openreach should "become truly open" by operating as a mutual body.
Last month, Ofcom published its Strategic Review of Digital Communication, which outlines a strategy to promote the rollout of ultrafast broadband facilities on a large scale, based on cable and fibre lines.
While the regulator stopped short of demanding that Openreach be separated from BT, it did suggest that it be governed "at arm's length from BT, with greater independence in taking its own decisions on budget, investment and strategy".
However, the Co-operative Party believes this decision means Openreach will continue to be run in the interests of BT's shareholders, rather than those who use phone and broadband services.
General Secretary Claire McCarthy has therefore suggested that Openreach should become a mutual organisation and be owned by the "households and businesses who rely on it".
Speaking to the Guardian, she said: "Run on a not-for-profit basis, a public-service Openreach would be tasked with ensuring a fair playing field for all providers that use its infrastructure.
"The proceeds from the services it provides would then be reinvested, pushing up standards and speeds for all."
Ms McCarthy went on to stress that fast and reliable telecoms are too important a part of the UK's economic future for "any single private interest to have a stranglehold on the underlying infrastructure".
While Ofcom is not advocating a complete split of BT and Openreach at the moment, the regulator confirmed it "remains an option".
One of its key recommendations is to support investment by rival providers to make the market more competitive and reduce the country's reliance on Openreach.
Ofcom hopes to achieve this goal by improving access to Openreach’s network of telegraph poles and its ducts - the underground tubes that carry telecoms cables. This will enable competitors to connect their own fibre optic cables directly to homes.