The committee also wants energy company Vattenfall to remain wholly state-owned.
The Social Democrats, the Green Party, the Left Party and the Sweden Democrats are behind the proposal, while the four parties of the centre-right Alliance government all abstained, according to a statement from the committee.
The Swedish parliament had previously called on the government to reduce state ownership in the three companies.
The current proposal seeks to revoke the measure.
The committee's majority argued that the state should use its SBAB shares to increase competition in the banking market, which they believe would benefit consumers.
In addition, the committee does not want the government to list Posten Norden on the stock exchange
The committee also emphasised that TeliaSonera owns a majority of the nation's IT infrastructure, including the network off access points, and suggested that the government instead devise proposals on how to separate the access network from TeliaSonera's other activities.
Some members of the Committee on Industry and Trade also sharply disagree with parts of the government's proposed budget, which mentions it may become necessary to consider selling Vattenfall shares.
“One argument is that Vattenfall manages key natural resources,” the committee said in a statement.
“Every year, Vattenfall brings a big contribution to the state's coffers," Lars Johansson, spokesperson for the Social Democratic Party, said in a statement.
"Properly managed, Vattenfall can be an extremely powerful tool for energy policy and can strengthen Swedish competitiveness through a more rapid energy adaptation."
The Moderates, Liberals (Folkpartiet), Centre Party and Christian Democratic party disagree with the proposal.
“State ownership of firms operating in a competitive market should be reduced or withdrawn,” stated the parties in a press release.
“The state's role as a legislator would thus be more streamlined, while promoting healthy competition.”
Revenues from the sale of shares would contribute to reducing government debt, they argued.