Speaking to Aljazeera on Friday, Kalugin said "the Russian side would urge Hamas to negotiate with Israel on the basis of previous agreements".
Vladimir Putin, the Russian president, made an unexpected move on Thursday in announcing he was planning to invite Hamas leaders to Moscow for talks in a bid to deter a fresh conflict in the region.
The movement won a landslide victory in Palestine's general elections in January.
Condoleezza Rice, the US secretary of State, won assurances from Russia on Friday that Moscow would send a firm message to Hamas that the resistance group must change if it wants world support, the State Department said.
Rice telephoned Sergei Lavrov, the Russian foreign minister, to discuss Moscow's surprise plans to invite Hamas for talks, a move that irked Washington and was not discussed when Rice met Lavrov in London last week.
"We have been assured that should the Russian government
meet with Hamas, that they would send that - that the meeting would be with the intent of sending that clear, strong message," Sean McCormack, the State Department spokesman, said.
An Israeli cabinet minister on Friday accused Putin of "stabbing Israel in the back".
The minister, Meir Sheetrit, also expressed concern that international pressure on Hamas is softening after France said it supported the Russian initiative.
In an AP interview, Sheetrit of the centrist Kadima Party called Putin's remarks an "international scandal".
Hamas won a landslide victory
in the January elections
"(Putin), I believe, would feel very bad if Israel would invite the Chechen organisations of terror into Israel and give them legitimacy," Sheetrit said.
Sheetrit, speaking to Channel 10 TV late on Friday, expressed fear that the French response was a "serious crack" in international pressure against Hamas. "Perhaps they are following Russia," he said. "This is totally unacceptable."
On Thursday Putin said at a news conference in Madrid: "Today, we must recognise that Hamas came to power via legitimate, democratic elections in the Palestinian National Authority, which is why we must respect the choice of the Palestinian people."
Kalugin said earlier that Moscow was seeking to convince Hamas to recognise Israel and observe existing agreements between Palestine and Israel and the road map peace initiative proposed by the UN, the US, the European Union and Russia - the international mediators in the peace process.
Responding to Putin's comments, Hamas leader Ismail Haniyah from Gaza was quoted as saying that the group's members would be "delighted" to visit Russia if they were officially invited.
Meanwhile, Russian Defence Minister and Deputy Prime Minister Sergei Ivanov said on Friday that many countries would be forced to maintain relations with Hamas eventually.
Ivanov said: "Many leading countries - including the countries of the 'quartet' - will come into contact with Hamas sooner or later, some way or another."