Minister for Health Simon Harris is likely to consult with his officials tomorrow to see how quickly legislation could be published to address the referendum result.
However, while there is an outside chance the legislation will be ready for publication before the summer break, it is more likely to be published during the Dáil recess in late July or August.
It is possible the bill could move very quickly through the Oireachtas if as expected it goes before the Dáil and Seanad in September.
The opposition has said that the legislation should be swiftly published in the coming weeks.
It is anticipated Mr Harris will meet with opposition leaders next week to discuss the issue.
As well as the legislation, clinical guidelines will have to be drawn up in conjunction with the new laws on abortion.
The regulation of new medical products, such as the abortion pill, will also have to be completed before the new law is passed.
There are calls for the Dáil to hold a special sitting over the summer to ensure the speedy passage of the legislation to repeal the Eighth Amendment to the Constitution.
At a press conference this morning, the Together for Yes group called on the Government to start the process of legislating for abortion next week.
The group's co-director, Orla O’Connor, said TDs and senators should consider a special sitting over the summer to ensure the passage of legislation.
Ms O'Connor said she would support the new legislation being called 'Savita's Law'.
The vote to change the constitution to repeal the Eighth Amendment, paving the way for new legislation to allow for the termination of pregnancies, was decisive with the Yes vote at 66.4% with 33.6% voting No.
Just one constituency, Donegal, rejected the proposal.
The country voted to repeal the Eighth Amendment by a majority of 706,349 votes.
The Eighth Amendment, which was inserted into the constitution in 1983, recognised the equal right to life of the mother and the unborn, and limited the circumstances whereby abortions could be carried out.
Ms O'Connor said that today women are still planning their journeys abroad to terminate pregnancies.
She said the "phenomenal" vote showed "the people have spoken and they've spoken very loudly" and that was why politicians needed to move quickly to introduce legislation.
Speaking on RTÉ's This Week, Minister for Employment Affairs and Social Protection Regina Doherty said the suggestion for the Dáil to sit in an extended session in order to pass the new law on the termination of pregnancy is not a bad idea.
She said the General Scheme of the Bill is at a very advanced level and would be much more specific than politicians would normally have in the Heads of a Bill.
She said the clear mandate from the people following the referendum is to do this and to do it fast.
"The sooner we do it, the better because there are women travelling [to have abortions] now and everyday until the law changes," she said.
Ms Doherty said she is heartened to hear politicians who campaigned against repealing the Eighth Amendment say that they will engage constructively and not filibuster the legislation.
Speaking on the same programme, Fianna Fáil TD Timmy Dooley said there is no reason why this legislation should not be published in the next month to six weeks, and politicians can have a fairly straightforward debate after the summer recess and have the law passed well in advance of the Budget.
Mr Dooley said he believes there will be a majority of his party colleagues who will vote in favour of the legislation that will be introduced, despite some of them having campaigned against repeal.
He said he would have "no issue" with the Dáil sitting in the summer to pass the legislation if the Government is prepared to arrange time.
"I certainly think it would be the right thing to do to get it done".
The Taoiseach has received calls from international leaders commending the decision by the Irish people following the referendum.
Among those to speak to Leo Varadkar were British Prime Minister Teresa May and Candian Prime Minister Justin Trudeau Xavier, as well as the leaders of Luxembourg and Estonia.
Speaking on RTÉ's The Week In Politics, Minister for Children Katherine Zappone said Mr Harris texted her this morning to tell her that he will seek permission from Government this week to begin drafting a legal framework around abortion law following the referendum result.
Ms Zappone said she would like to see the legislation on the floor of the Dáil before the summer break but conceded this may not be possible.
She also said that she had discussed the issue with Attorney General Seamus Woulfe who advised that it could be "quite challenging to complete" the bill.
Independent TD Michael Fitzmaurice said while he was surprised by the overwhelming Yes vote in the referendum, he thinks parliamentarians who wanted a No vote have to respect the will of the people
Speaking on the same programme, he said he does not believe that the legislation will be obstructed in the Dáil.
"I believe each parliamentarian will have their view - this is a referendum. People have gone to the polls. People have voted. I cannot see it being obstructed in the Dáil. There will be people, like myself who have deep reservations on the 12 weeks, but I personally will not be voting against something."
Speaking on RTÉ's This Week, Sinn Féin leader Mary Lou McDonald said the referendum result means "a case for change, for protecting our health and our lives is now unanswerable".
Elsewhere, the National Association of General Practitioners has said it is to call an emergency meeting shortly of its members to discuss the outcome of the referendum.
The NAGP said the meeting will consider what impact the decision might have on GPs and their patients.
In a statement it said the organisation had previously highlighted concerns it had in the lead up to the referendum, that an abortion service would be GP-led.
The NAGP said it is now seeking engagement with the Department of Health "to devise a respectful, safe and supportive pathway for women in crisis pregnancies."
It added that it would always be up to each individual GP to personally decide if they wished to provide any service.
Additional Reporting Samantha Libreri