Taoiseach Leo Varadkar has said the Government is hopeful that schools may be able to reopen in May and June.
Mr Varadkar also said: "there is a very good chance" that the Leaving Certificate exams will go ahead and for that reason he would encourage students to continue studying for them.
He said the Government was doing "everything possible or feasible that we can so that that group of young people could start college as normal in October".
Earlier, Minister for Education Joe McHugh confirmed that schools would not be reopening in the short term as efforts continue to stop the spread of Covid-19.
In relation to state exmainations Mr McHugh said that he and his department "really want to make those exams happen".
He said they owed it to young people to "see this through and try to have the exams this year".
He added by that he meant "the dates that are defined for this year" and said his department was working towards this.
The Taoiseach announced on Thursday 12 March that all schools, colleges and childcare facilities would close in response to the coronavirus pandemic.
Today, as the Government was announcing a series of significant new restrictions to deal with the virus outbreak, the Taoiseach said: "We are all hopeful that schools might be able to come back in May and June and to have the Leaving Cert in the normal way", the Taoiseach said, but he added that if that did not happen then the Minister for Education was working on contingency plans to allow the Leaving Cert to go ahead.
"The Junior Cert also if possible", he added but said obviously the Leaving Cert was more important.
Minister McHugh acknowledged that this "is an extremely difficult time for everyone, not least students and their parents".
In a statement issued this evening he said his message to students facing exams is that they should keep focused, keep working and try, as much as possible, to prepare as normal for the state exams.
"We are doing everything in our power to make sure those exams happen", he said.
The minister praised teachers and students who "have answered the call to remote learning with exceptional flexibility and adaptability".
He said he was deeply conscious of significant work being done to ensure continuity of learning across higher and further education institutions.
There have been 1,125 cases of coronavirus confirmed in the State and six people with Covid-19 have died.
More coronavirus stories
The Teachers Union of Ireland has urged students due to sit Leaving and Junior certificate exams this summer to ignore speculation - particularly on social media - and continue to engage wholeheartedly with their studies.
The union's president, Seamus Lahart said students could "take it as a given" that every effort was being made "by the State Examinations Commission and by everybody in the broader education community to have appropriate arrangements in place for their examinations".
He said the TUI appreciated that the decision to extend the school closures was a necessary one and that educators would continue to do their work to the highest professional and ethical standards.
Meanwhile, Maynooth University has said that it will not resume face-to-face teaching this semester.
The university said that "remote teaching will continue until the end of the semester. Exams will be replaced with equivalent remote assessments".
It added that university operations and research will continue with many staff working from home.
"MU residences remain open. Students who wish to stay until the end of their lease are welcome to do so. Students who decide to vacate their university accommodation will have their rent refunded. The previous policy being circulated reflected the University's standard policy prior to the Covid-19 outbreak."
Youth Work Ireland, which represents youth services across the country, called for clarity on the Leaving and Junior Certificate June exams for young people and their parents.
It said the most disadvantaged are missing out because they have less support at home to do educational work independently.
The organisation said it believed there was growing uncertainty about the feasibility of holding state exams. In a statement it pointed to the UK decision to cancel A levels.
Michael McLoughlin of Youth Work Ireland said key decisions about third level and other career options depended on the exams.
"If alternative arrangements are to be put in place a lot of work will have to be done, students may effectively be required to repeat a year and backlogs or congestion could arise for future years", he said.
He called for young people and parents to be involved in coming up with any solution.
"A lot of thought, planning and discussion is needed around this question and young people and parents need to be involved. There may be ways of delivering the Leaving Cert only but increasingly some students are going to be disadvantaged compared to others."