The select committee has twice written to the ambassador asking him to front, so MPs can ask questions about Russia's invasion of Ukraine.
Similar requests were made to the Ukrainian and European Union ambassadors. They both accepted the invitations and spoke to the select committee about the invasion last month.
RNZ has been told the Russian ambassador declined the select committee's first request.
MPs then wrote to the ambassador a second time, asking again for him to appear. In that letter, RNZ understands the committee outlined the powers Parliament has to compel someone to appear.
The ambassador has not yet responded to that letter.
RNZ understands the select committee is actively investigating if it can summons the ambassador.
Under the Vienna Convention, however, the ambassador has diplomatic immunity, which means he isn't obliged to give evidence.
RNZ understands MPs have sought advice about whether it is still possible to haul the ambassador in for questioning despite this and how it could be done.
The Foreign Affairs Select Committee does not have the power to summons a person.
It would have to apply in writing to the Speaker of the House, who would then decide if the person should be sent for.
It is extremely rare for a person to be summonsed to appear at Select Committee. In fact, it has only happened once in living memory.
In June 1996, the Justice and Law Reform Committee issued three summons for members of the Road Knights gang as part of an inquiry into gang activity.
Only one gang member was served but never showed up, so the matter was dropped.
The approach from Parliament is in line with the Government's condemnation of Russia's actions.
Days before the invasion, the government called in the Russian ambassador to "hear New Zealand's strong opposition" to Russia's escalating actions.
The government has since actively matched sanctions imposed by Western countries, introducing sanctions against Russian oligarchs, officials and their families.
It has also imposed 35 per cent tariffs on all Russian imports, contributed $11 million in humanitarian aid and non-lethal military assistance to support Ukraine and shipped surplus defence equipment to Ukraine's military.
Unlike Western countries like the UK and US, New Zealand has not sent weapons to Ukraine.
The Defence Force has offered to send javelin missiles, but Cabinet is yet to take up the offer.
The Russian Embassy did not respond to RNZ's request for an interview.
RNZ has made repeated requests to speak to the Russian ambassador since the start of the invasion, but all have been either declined or ignored.
Green Party foreign affairs spokesperson Golriz Ghahraman said the ambassador should be expelled if he isn't willing to engage with MPs.
"The Russian ambassador continues to be in New Zealand with the promise that this will keep dialogue open, so if that promise isn't actually being delivered on ... then the position should be that he is expelled. Because we are keeping him here in a way that legitimise Putin's regime."
Until now, the Green Party has not called for the ambassador's expulsion.
It joins both National and Act, which have been urging the government to minimise its diplomatic ties with Russia.
"We haven't called for the ambassador to be expelled, because like any other diplomats, the work of negotiating and speaking to peace, and furthering the interests of victims that suffer from this war, is the priority, not some kind of political grandstanding.
"But we haven't so far had an ambassador that has been overtly obstructive of that job. The job of a diplomat is to continue dialogue, so that's what we honour until he ... honours the same," Ghahraman said.
Prime Minister Jacinda Ardern has defended the decision to allow the ambassador's continued presence, saying the government wants to keep consular and diplomatic channels open.
It is likely Russia would expel New Zealand's ambassador if the Government kicked the Russian diplomat out.
As of last week, according to the Ministry of Foreign Affairs, eight New Zealanders in Russia, including individuals and families, have sought consular support or advice in the last month, with the embassy in Moscow also receiving a small number of enquiries and requests for information.
Ardern has not ruled out expelling the ambassador.