The network, while actively seeking a change in the law, is shrouded in a cloak of secrecy with the group's founder, Eva Flyborg, unwilling to divulge the identity of members.
"The atmosphere is such that we do not want to tell you how many we are or who is involved. Euthanasia is an issue which stirs up a lot of emotion," Flyborg told the Riksdag & Departement newspaper.
Flyborg told the newspaper that the network includes representatives from "in principle" all of the parliamentary parties.
Flyborg, together with Frederick Federley and Staffan Danielsson of the Centre Party, and Helena Leander of the Green Party, have penned a motion to call on the government to act for a change in legislation outlawing the practice.
"It is not worthy of Sweden in the 2000s for seriously ill people to be tormented and tortured to death," Flyborg argued, claiming that "between 80 and 90 percent" of the Swedish population support the idea of being of control of the last days of one's life.
Flyborg explained that the issue of euthanasia remains controversial in Sweden and compared the debate to when abortion was discussed in the 1970s.
She characterised the debate as a division between religious groups arguing for the sanctity of God's creation and those arguing for the right to one's own body.
Flyborg argued that it was time for Sweden's politicians to catch up with the Swedish people and act on the issue.
The issue has received a great deal of media attention recently in connection with the trial of a 57-year-old doctor for the manslaughter of a three-month-old baby that was born prematurely and was expected to die shortly of brain damage.
The case is exceptional because legal procedures against doctors are very rare in Sweden, where malpractice cases usually go before the National Board of Health and Welfare (Socialstyrelsen) with no criminal ramifications.
Furthermore it has raised concerns in medical circles, leaving doctors uncertain and wary of performing certain operations for fear of legal consequences.