The SVP currently has one of the government seats, held by Ueli Maurer, defence minister, although it holds more than a quarter of the seats in the upper house of parliament.
Felix Müri, an SVP MP from the canton of Lucerne who is vice-president of the party's parliamentary group, made waves on Monday by saying the party is prepared to go into opposition if it fails to gain an extra seat in the government.
The SVP, a nationalist party which favours restrictions on immigration, has long been the most popular party in Switzerland, although it has been opposed by the other parties in the Swiss multi-party political system.
In 2007, it lost one of its two seats in the government when a moderate member of the party, Eveline Widmer-Schlumpf, agreed to seek election against the SVP's Christoph Blocher, who was then an incumbent government minister.
Widmer-Schlumpf was elected but soon became a member of a new party, the Conservative Democratic Party (BDP), after she was booted out of the SVP.
She was subsequently re-elected to government in 2011 as a member of the BDP, leaving the SVP again with only one seat.
Government members are voted in by a joint assembly of the upper and lower houses of parliament following the federal elections held every four years.
Traditionally, the main parties agree to a coalition government with balanced representation from the left, right and centre.
So elections for government typically involve a lot of horse trading among parties.
But Müri said many SVP members are “fed up” with the current situation, which leaves the SVP under-represented in the government.
“If we do not get the second seat at the expense of the BDP, we will have Ueli Maurer withdraw from the Federal Council (government) and go into total opposition,” he told the Blick newspaper.
By tradition, government members do not voice disagreements publicly and always present a united front once government decisions have been made.
Going into opposition would allow the SVP to voice its views freely and to launch referendums on key initiatives.
Already, the SVP successfully spearheaded an initiative, approved by voters in February 2014 but opposed by the government, to curb immigration from the European Union.
Blick said the SVP is already showing traits of an opposition party “but the tone would worsen dramatically if the largest party were no longer represented in the government”.
Pascal Sciarini, a political scientist at the University of Geneva, said the SVP has actually been able to gain popular support by not having two members in the government.
“With one foot in government and the other foot outside government it is very profitable for (the party) electorally,” he told Le Temps newspaper.
Sciarini said the system of electing the Swiss government, once referred to as the “magic formula” because of the way it balanced representation for the major parties, has changed.
Neither the “magic formula” nor a mathematical system of power sharing now exists but rather a system of “fluctuating consensus,” he said.
Swiss voters go to the polls on Sunday to elect MPs and members of the council of states (the senate, or upper house of parliament).
Polls show the SVP as the most popular party for the lower house but a proportional voting system means that seats will be allocated among several parties — 11 are currently represented.
By contrast, in the 46-seat senate, where members are voted by majority in each of the country's 26 cantons, the SVP only has six seats.