Greek Finance Minister Yanis Varoufakis on Tuesday said Greece needs “a bridge agreement” on its debt situation that could in turn lead to a full accord by June.

Greece’s new Prime Minister Alexis Tsipras and Mr. Varoufakis are touring European capitals in an effort to persuade fellow European leaders to reverse the stringent terms of Greece’s EUR240 billion ($272 billion) bailout.

“What is needed is a bridge agreement that gives us some time–for example a month or six weeks starting from late February–to find an agreement that we would then put in place starting from June 1,” Mr. Varoufakis said.

Such an agreement is possible “provided that in Europe everyone calms down,” Mr. Varoufakis added. The minister was speaking with Italian journalists during a visit to Rome to meet Italian officials.

After a meeting with Italian Economy Minister Pier Carlo Padoan, Mr. Varoufakis declined to provide any further details of Athens’s proposal for its debt beyond saying that Greece is working on a “road plan” for a solution.

“We could envision an end to the Greek crisis starting from June,” he said. “You’ll see. If there’s an agreement, capital will start flowing very quickly.”

Mr. Varoufakis will meet with European Central Bank President Mario Draghi in Frankfurt on Wednesday and plans to meet on Thursday in Berlin with German Finance Minister Wolfgang Schäuble, for whom he expressed “great respect.”

“I only entered politics three weeks ago,” the Greek minister said on Tuesday. “Therefore, I remain optimistic on the possibility of an agreement.”

Mr. Tsipras was scheduled to meet Italian Prime Minister Matteo Renzi later Tuesday afternoon, after which the pair are scheduled to hold a news conference.

Mr. Varoufakis’s comments came after European Commission President Jean-Claude Juncker signaled on Tuesday that there is limited room for maneuver to accommodate the demands of the new Greek government.

The Greek premier is due in Brussels on Wednesday morning to meet Mr. Juncker among others. It is the Greek leader’s first trip to Brussels since taking office last week. He is also visiting Paris this week as he seeks to drum up support for his government’s push for greater leniency on its debt repayments.

“I have already said to him twice on the telephone that we have to take account of the democratic expression of the Greek people,” Mr. Juncker told lawmakers in the European Parliament, referring to the Greek leader. “But I think those who won the elections will also have to take account of the way in which others do things and the convictions of others.”

“So yes, we are going to have to do a certain amount, but we aren’t going to change everything because of one electoral result that some people like and some people don’t like,” he said.

Mr. Juncker said that while the EU executive hasn’t always got the balance right between fiscal consolidation and growth, those suggesting there were no real constraints on the bloc’s budgetary position weren’t credible.

“What we have done over recent years hasn’t necessarily always taken account of growth…So I do think we have to correct the errors we have made in the past,” Mr. Juncker said. “What I don’t think we should do is simply replace what we have already had with its exact opposite. That will basically send us to the wall.”

Greek stocks and bonds rose earlier Tuesday on a series of comments to the media by Mr. Varoufakis suggesting that Athens may propose trading Greek government debt held by the ECB for growth-linked bonds. That would possibly lower the risk of losses to privately help bonds.

Those comments left investors with the view that the Tsipras government is backing off from tough demands regarding the terms of the country’s bailout. The new government also wants less oversight by inspectors from the so-called troika–the EU, the ECB and the International Monetary Fund.