The meeting - one of about a dozen Ms Bishop held with various foreign diplomats - came after Mr Churkin earlier vowed that Russia, as one of five permanent members of the Security Council, would veto the resolution.
"We will vote against. I have no doubt in that," Mr Churkin told the Russian TASS news agency.
Russia last week presented the Security Council with its own draft resolution, which did not include an international criminal tribunal.
The move followed the leaking of a report by a Dutch-led investigation that supports claims Russian-backed separatists were responsible for the downing of the Malaysia Airlines plane, shot down over eastern Ukraine on July 17 last year an hour into its flight from Amsterdam to Kuala Lumpur.
All 298 passengers and crew, including 39 Australian citizens and residents, were killed.
But Ms Bishop said it was only logical that Russia should support the tribunal, after having last year backed resolution 2166, which had called for "a full, thorough and independent international investigation".
"I did remind the Russian representative today that Russia had backed the ceasefire, the retrieval of the bodies, the investigations, and so of course it was logical that it should back the setting up of the tribunal which would be able to investigate the causes of the crash and to hold those responsible to account," Ms Bishop said.
The resolution would pass with the support of nine of the 15 Security Council members, but can be vetoed by Russia as well as the four other permanent members - China, the United States, the United Kingdom and France.
"I'm not sure whether we will get all of the 15 members on board. I'm optimistic but there's always a chance of a veto," Ms Bishop said.