The two European countries share a border straddled by the High Tatras mountains, a favourite winter holiday destination for locals and tourists from Russia and Ukraine.
Fico met representatives of the Slovak Olympic Committee, the High Tatras regions and the national ice hockey and ski associations in the northern town of Poprad, which unsuccessfully bid for the 2002 and 2006 Winter Games, to debate the plan.
"The fundamental question is whether we can afford it or not, whether we have the needed infrastructure for disciplines expected to take place here," Slovak news agency TASR cited Fico as saying.
Poland's Sports Ministry has said it would back a proposal to co-host the Winter Olympics and the government would soon discuss providing the necessary guarantees for the bid.
The Slovak government is expected to announce its formal support for the bid by the end of February.
Poland successfully co-hosted last year's Euro championship with Ukraine.
Fico said he envisaged Slovakia hosting ice hockey and skiing events. The cost of hosting the Games would be shared between the International Olympic Committee (IOC), which was expected to provide some 40 percent of the funds, and the two countries with Poland providing the larger share.
The IOC has set November 2013 as the application deadline and the host city will be elected in July 2015.
Fico, who plays soccer and goes running, said Slovakia should not be over-ambitious about the plans.
"We mustn't have big eyes," he said. "It would be a great success should we succeed with Poland, the candidacy alone will mean further PR for Slovakia and it will be a great opportunity for development of the infrastructure."