"Our discussion was constructive despite our divergent positions," Slovenian Prime Minister Miro Cerar said of his first talks with counterpart Andrej Plenkovic since the June 29 ruling.
The dispute over where the two countries' sea borders should be drawn has poisoned relations since both declared independence from Yugoslavia in 1991.
The international arbitration court recommended that Slovenia should have uninterrupted access to international waters, allowing "freedom of communication" to all ships seeking access to the small country.
It also awarded Slovenia a larger stake of the Piran bay -- currently divided in half by Slovenia and Croatia -- and ruled on many tiny disputes along their 670-kilometre (415-mile) land border.
The ruling -- based on an EU-backed agreement signed between the two countries in 2009 -- was rejected by Zagreb which said it was not binding.
Cerar said Wednesday he hoped Croatia would relent and that the two EU members would not have to take up an offer from the European Commission to help facilitate the process.
Plenkovic told a joint news conference said that Brussels "has no competence in the drawing of borders. European law is not a substitute for international law."
"Croatia wants to put on the table, through an open dialogue, all the border-related issues, and to search a solution that will be acceptable for both sides," Plenkovic said.
He invited Cerar to visit Zagreb in September for further discussions.
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