The SNB set the cap last September to try and stave off the risk of deflation after safe-haven buying from the euro zone pushed the franc up sharply. The central bank has repeatedly vowed its determination to defend the level by buying unlimited amounts of foreign currency.
But as the euro zone crisis escalates, concern has mounted that the cap may come under pressure from huge capital inflows, leading to debate about what instruments the SNB could use to fight the strength of the franc if the euro were to collapse.
"Ahead of the SNB's quarterly Monetary Policy Assessment in mid-June, the discussion on the lower boundary in EURCHF is back," UBS economist Reto Huenerwadel said in a note.
"The question to answer thereby is not so much on whether the SNB can, but whether they want to support the lower boundary at 1.20 in light of lingering risks both domestically and internationally and ultimately their stance on inflation."
In a weekend newspaper interview, former UBS Chief Executive Oswald Gruebel said the SNB should give the cap up to shield the economy.
Some Swiss politicians said separately that the SNB should consider capital controls and introduce negative interest rates.
But Pascal Gentinetta, director of business group economiesuisse, said blocking inflows of funds would take a huge effort and have no guarantee of success.
Switzerland's economy minister Johann Schneider-Ammann said imposing negative interest rates would have a limited impact.
Despite fears that the Swiss economy would slip into recession, it has performed well due to strong domestic consumption.
The franc dipped 0.2 percent against the dollar compared to Friday's New York close to trade at 0.9674 by 0619 GMT.
The franc was steady against the euro at 1.2008.