Athens objects to its northern neighbour's name arguing it suggests that Macedonia has claims to the territory and heritage of Greece's historic northern region of the same name.
The dispute has remained unresolved since the former Yugoslav republic's independence in 1991.
In light of the Greek objections, the country joined the United Nations in 1993 as the Former Yugoslav Republic of Macedonia (FYROM).
But Macedonia has made significant progress recently and the European Union has said it is "very confident" the name row will be settled before July.
Skopje in February made a concession to reconcile with Athens by renaming the capital's Alexander the Great airport as Skopje International Airport, which paved the way for a visit by Greek Foreign Minister Nikos Kotzias.
Kotzias arrived in Macedonia on Thursday, on board the first Greek plane to land at Skopje airport in 12 years.
"We believe in an honourable compromise, to the benefit of both sides," Kotzias told a press conference after meeting his Macedonian counterpart Nikola Dmitrov, according to Greek media.
"In an honourable compromise one cannot have all one wants," added Kotzias whose visit comes a week ahead of a fresh round of UN-mediated talks over the issue in Vienna.
"We have to overcome the current difference (over the name) ... which is not easy," Dimitrov said.
He called for a "common solution" that would be "worthy ... without humiliation."
The compromise solutions, cited by media of the two countries, consist of adding an extra geographic designation to Macedonia's name such as Upper Macedonia, New Macedonia or Northern Macedonia.
But in both countries the governments are facing the opposition of nationalists who refuse any concession over the issue.
In Skopje Friday a small group of some 30 protesters opposed to any name change rallied across from the foreign ministry waving Macedonia's flags.
A resolution of the name issue is needed before Macedonia, home to some 2.2 million people, can join NATO or the European Union.
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