"Perhaps it is better to leave the decision up to the next government," Finnish Justice Minister Anna-Maija Henriksson was quoted by Finnish national broadcaster Yle as saying.
Saturday was the last working day for the current parliament, which will convene again after the general elections in April.
Among the 185 ILO member states, only 20 have ratified the Convention since it entered into force in 1991. The convention aims to enhancing the rights of indigenous people.
The Samis, whose population amount to 6,000 in Finland, believes an ILO-level recognition will give them further influence in relevant matters.
The Finnish government once tried to introduce a concept of "identification by a group" which would have given the Sami community right to decide who is a Sami and who is not. But parliamentary majority wanted to retain the current ruling under which a person can choose whether he or she is a Sami or not. All members of parliament from the Lapland area were in favour of retaining the current definition.
Young Samis in the Helsinki area demonstrated outside Parliament building on Friday. "We ourselves must have the right to decide who is Sami and who is not", said protester Anna-Maria Magga.
Camilla Busck-Nielsen, a civil servant in charge of Sami affairs at the education ministry, told Xinhua that the Samis enjoy full social rights and duties in Finland.
Samis have extensive public radio and TV services of their own despite the relatively small population. News in Sami area is seen nation-wide on national TV.
The Samis fell under the dominance of the then Sweden from early medieval period and their shamanic religion was discouraged.
After Finnish independence in 1917, Finland pursued assimilation policies and authorities discouraged the use of the Sami languages. Policies towards the Samis began to change in the 1970s.
Today the law requires primary education to be given in Sami languages. Secondary level courses in Sami language are available in areas inhabited by ethnic minorities.
Only a half of the Samis in Finland speak the Sami language as a native tongue. There are Sami inhabitants also in Sweden, Norway and Russia.