Irish Prime Minister Bertie Ahern also plans to issue a report on the constitution on Thursday evening at the Brussels meeting, when he will push for formal resumption of talks.
An attempt at a first EU constitution failed in December when EU member Spain and incoming member Poland refused to give up generous voting rights allocated to them under a 2000 treaty.
This was despite fears that the bloc - which expands from 15 to 25 countries in May - will become unworkable without a common basic law.
But the door to negotiations has reopened, after the election defeat of Spain's conservative government on 14 March. The incoming Socialist Party has pledged to work towards agreement on the EU constitution.
Addressing a news conference in Dublin late on Monday, Ahern said he was encouraged by the contacts he had had with his EU counterparts, particularly a meeting with French President Jacques Chirac in Paris earlier in the day.
"I have been heartened by the support I have received in recent weeks and believe that it is possible we can conclude the work of the Convention by the June Council," he said.
He said his recent meetings with government leaders, including those of the 10 nations scheduled to enter the EU, also went extremely well.
"There is now a growing support that we finish the constitution by the June council"
Irish prime minister
"I think there is a clear commitment by people to try to work with us to make progress during the Irish presidency. If the meeting on Thursday night goes well, hopefully we will be able to continue to progress."
Ahern and Chirac said in Paris earlier Monday they saw a possible deal emerging on the stalled draft constitution by the end of June.
Chirac said he was determined to do everything he could to achieve agreement on the document before an EU summit on 17-18 June, in the final weeks of Ireland's six-month presidency of the bloc.
"We think it is possible to reach a conclusion by that date," Chirac told a joint news conference with Ahern.
The Irish prime minister agreed such a deal was possible.
"There is now a growing support that we finish the constitution by the June council, we have to decide whether we do it then or before then but we will continue to work with our colleagues."