The ASTI has said it will re-enter discussions with the Department of Education on assessment for this year's Leaving Certificate.
The union withdrew from talks on Thursday citing unhappiness with the direction the talks had taken.
In a statement, the union said that following an unequivocal statement of commitment from the Minister for Education to holding the Leaving Certificate exams this year, it believes that doubts regarding the staging of additional component elements of this year's exams have been substantially alleviated.
It said it had also been assured that teachers will not be required to provide a list ranking their students in order of merit, like they did last year.
The ASTI said that at engagements with Minister Norma Foley and her officials since its withdrawal from talks, it raised its concerns about the staging of additional component elements of this year's exams, such as coursework, orals, and practical exams, and the consequent relegation of the Leaving Certificate to what it described as a secondary position.
The union said it also raised the issue of a lack of data available in many schools to base and inform the parallel or corresponding measure that the minister is seeking to put in place alongside the Leaving Certificate.
It said: "In addition, the ASTI has been assured that the requirement for teachers to provide a list of their students in rank order of merit, like they did last year, will not feature in any corresponding/parallel measure that may be implemented this year."
The unions said on the basis of all this it would now re-enter the discussions.
"We will continue to constructively engage to seek to ensure ways are found to facilitate this year’s Leaving Certificate students in moving on with their lives", it concluded.
In a statement, the department said that Minister Foley welcomed the ASTI's return to the talks and "notes their willingness to engage on the agenda that has already been set out in the process".
It added: "The minister wishes to thank the TUI, parent, student and managerial bodies for their constructive engagement which continues over the weekend as the issues continue to be worked out.
"Intensive engagement is continuing with the education stakeholders to advance progress and provide certainty and clarity at the earliest possible time."
The Teachers' Union of Ireland (TUI), which was also involved in the talks, said it noted the "decision of the ASTI to rejoin the Leaving Certificate talks based on information and assurances that the TUI had already sought and secured".
It added : "At every point the TUI has insisted upon and received assurances that the Leaving Certificate written examinations would take place subject only to public health advice.
"It was on this basis that we entered the confidential talks process. One of our key priorities, to ensure that the other components of assessment would also take place, had been significantly advanced following detailed proposals by the TUI."
Calls for stakeholders to work together
Today, Minister of State for Local Government Peter Burke said the Government was prioritising the reopening of schools "over everything else".
He said certainty cannot be given at every step on the reopening due to Covid-19.
Speaking on RTÉ's Saturday with Katie Hannon programme, he said he shared students’ frustration about the situation they face, adding that all stakeholders now need to work together to bring clarity to students.
Speaking on the same programme, Sinn Féin TD Louise O’Reilly said all stakeholders need to come together to produce a clear statement that they will put students first.
She said it beholds all of those involved to give students choice now, as there has been a lot of time lost already.
Also speaking on the programme, Labour TD Aodhán Ó Ríordáin said it was now good that all stakeholders were back at the table.
However, he said he was taken aback by reports of "cast iron guarantees" by the minister on exams, when nobody could give that level of certainty yet.
Mr Ó Ríordáin said if a calculated grades process is returned to this year, then the profiling of schools needs to be ruled out.
Unions promise to hold minister to her word
Earlier, ASTI General Secretary Kieran Christie said the union will hold Minister Foley to her word.
He said that the union was now confident the Leaving Cert exams, with all of its components, will run this year.
"We are prepared to work with the minister on creative solutions that might work," he said.
Speaking on the same programme, Michael Gillespie of the TUI said he was "surprised" at the suggestion from the ASTI that anything had advanced regarding the situation.
On the issue of the ranking order lists, he said: "There were proposals on the table that were being worked on that we believe would form a solution that we could all live with this year."
Mr Gillespie said the TUI brought "robust proposals" to the talks regarding second assessment components, such as oral exams.
He said he believed it was possible for Leaving Cert classes to return to school during the week of 22 February, alongside classes for children with additional needs.