The Alliance government has been split on the issue, with the junior partner Christian Democrats opposed to the use of the word "marriage" for homosexual unions.
However the three other parties, the conservative Moderates, the Liberals and the Centre Party, are in favour of a gender neutral law that eliminates the current reference to marriage as something between a man and a woman.
"Regardless of sexual orientation, people in stable couple relationships have a need to manifest their feelings and their desire to live together," the motion reads.
While heterosexuals in Sweden can choose to marry in either a civil ceremony or a church ceremony, homosexuals are only allowed to register their partnerships in a civil ceremony. But this could all change, with parliament likely to pass new legislation on the matter.
The opposition Social Democrats, the country's biggest party, also support such a law, and together the parties would garner enough support to adopt the legislation in parliament.
Civil unions granting gays and lesbians the same legal status as married couples have been allowed in Sweden since 1995.
If the new legislation is adopted, Sweden, already a pioneer in giving same-sex couples the right to adopt children, would become the first country in the world to allow gays to marry within a major church.
The Lutheran Church, which was separated from the state in 2000, has since January 2007 offered gays a religious blessing of their union. It has previously said it wants the word "marriage" reserved for heterosexual unions.
Pastors who do not want to perform a gay wedding ceremony may however have the right to refuse, something gay rights' activists have criticized.
In 2007, 74 percent of Swedes were members of the Lutheran Church.