Sinn Féin leader Mary Lou McDonald has called on the FAI not to participate in the UEFA Nations League matches against Israel and added "Israel should be given the red card".
Ireland has been drawn against Kosovo, Israel and Austria and in Group B3.
The two games between Ireland and Israel are to take place next September and October.
The away fixture will take place on 27 September, while the return leg in Dublin is scheduled for 4 October.
The Football Association of Ireland said in a statement yesterday that it intends to fulfil the fixtures.
Last year, the FAI voted to submit a motion to UEFA to ban Israel from European competitions.
The FAI are yet to confirm if the October game will take place in Dublin or at a neutral venue, a decision that does not have to be finalised until 120 days before the game takes place.
Ms McDonald said the FAI have been put in a "very difficult position with this draw".
However, she said "the truth is that these matches should not and cannot go ahead".
"We are all aware that UEFA is allowing Israel to compete in competition, even though they are breaking every rule in the book.
"Israel is engaging in genocidal actions in Gaza and Ireland has to take a stand. There's only one correct thing to do here, and that is not to host or not to participate in these games," Ms McDonald said.
"Israel should be given the red card. That should be absolutely clear, and Ireland should be to the forefront in confronting them for their actions, their genocidal actions in Palestine, not giving them any comfort whatsoever," she said.
Ms McDonald also called on broadcasters not to cover the matches if they go ahead.
"I'm calling on the FAI to boycott the matches, not to give cover to genocidal Israel, and I'm calling on broadcasters similarly, not to broadcast those matches.
"Israel needs to get the message loud and clear. Netanyahu needs to hear loud and clear that we are calling it, calling genocide, and there is nowhere for him to hide ... we will not be acquiescent in that in any way," she added.
Ms McDonald said she disagrees with the Government's stance on the issue.
"I'm alarmed, frankly, that the reflex of the government and the Taoiseach was simply to give a Bible to Israel," she said.
Mr Martin said a decision on where the away leg of the fixture should take place was a matter for the football authorities.
He said that security authorities in Ireland would make an assessment on safety for the home leg in good time before the fixture takes place to ensure that it could take place "in a secure environment".
"It should go ahead, and I think the FAI has taken the correct decision to fulfil the fixture," Mr Martin added.
'Outrageous' that Taoiseach said game should go ahead - Richard Boyd Barrett
Speaking on RTÉ Radio 1 Saturday with Cormac Ó hEadhra, People Before Profit TD Richard Boyd Barrett said that UEFA was breaking its own rules as it threw Russia out of the contest over the war in Ukraine but is not treating Israel the same.
He said that there was a failure at Government level to impose any sanctions on Israel since the conflict in the Middle East escalated.
"It was really outrageous that Micheál Martin said that the game should go ahead.
Speaking on the same programme, Minister of State Marian Harkin, said that it was challenging when politics and sport overlap and that there are double standards here, but it is up to the FAI to decide on the fixture.
She said that if it were up to her, she did not think it should go ahead and she would not attend the match personally.
She said that none of this is simple or straightforward and Ireland has shown it stands up for Palestinians.
Conor D McGuinness Sinn Féin TD said that Israel should not be in the competition and Ireland should not fulfil this fixture, rather Ireland should pass the Occupied Territories Bill.
He said that there should be a sporting and cultural boycott of Israel until they come back onside in terms of international law.
He added that 'it beggars' belief that the Taoiseach said the matches should go ahead. He said that Mr Martin was "behind the curve" and "utterly wrong on this".
