Macedonian lawmakers voted to rename their country the Republic of North Macedonia on Friday and the agreement now needs backing from the Greek parliament to come into effect.
However the proposal has been met with resistance by many in Greece, who object to its neighbour being called Macedonia because it has its own northern province of the same name.
Ahead of the vote in Athens, two news websites alleged that a prominent member of Greece's main opposition New Democracy party -- which rejects the name deal -- called on its members to phone MPs of the nationalist Independent Greeks (ANEL) party and urge them to vote against the agreement.
ANEL is the junior party in Greece's ruling coalition and strongly opposes the name change, a rift that threatens to destabilise the government of Prime Minister Alexis Tsipras, who helped broker the deal.
The New Democracy member allegedly revealed the personal mobile phone numbers of ANEL MPs.
Athens News Agency said the news reports prompted the preliminary investigation into the allegations.
The investigation, according to judicial sources, is looking into whether personal data was violated as part of the probe.
ANEL leader Panos Kammenos, who is Tsipras' defence minister, has threatened to pull out of the government when the deal comes to a vote, but some of his party's MPs remain ambivalent.
Tsipras will have a crunch meeting with his coalition partner on Sunday morning in order to reach a make or break decision ahead of the vote.
If Kammenos refuses to back down, Tsipras hopes to rely on lawmakers from the small pro-EU To Potami party to get the agreement approved.
Macedonian Prime Minister Zoran Zaev on Saturday called on Greece to ratify the deal.
"Our parliament found the strength but it wasn't easy. But I am convinced that the Greek parliament will also find the strength to make the decision," he said.