Key ministers – including some within the Prime Minister’s own Cabinet – have indicated their support for reform to resolve an "anomaly" within the British Isles.
Scores of MPs across the Commons have indicated they are prepared to act to rewrite the current legislation given the absence of a devolved administration in Stormont.
But Mrs May’s fragile administration depends on the support of the 10 Democratic Unionist Party MPs – who strongly oppose any reform to Northern Ireland's strict laws.
Mrs May has not publicly commented on the result of the Irish vote, but Downing Street is understood to believe that any reform "is an issue for Northern Ireland".
"It shows one of the important reasons we need a functioning executive back up and running," a source said.
Pressure from within her own party could force the Prime Minister to act, with Education Minister Anne Milton suggesting she would back liberalisation if there was a free vote.
The current situation "does feel anomalous", she told ITV's Peston on Sunday.
Cabinet minister Penny Mordaunt – who is responsible for the women and equalities brief in Government – said the referendum signalled a "historic and great day for Ireland" and a "hopeful one for Northern Ireland".
"That hope must be met," she added.
Four former holders of the women and equalities role – Amber Rudd, Justine Greening, Nicky Morgan and Maria Miller – all back Ms Mordaunt in support for reform in Northern Ireland, the Sunday Times reported.
Labour MP Stella Creasy claimed more than 140 parliamentarians had already signalled support for an effort to change the law in Northern Ireland.
In a message to the DUP, she said the people of Northern Ireland "consistently support change" in the abortion law and it was "time to put them, not power in Westminster, first".
But DUP MP Ian Paisley said Northern Ireland "should not be bullied into accepting abortion on demand".
"The settled will of the people has been to afford protections to the unborn life and protect the life of the mother," he said.
Abortions are currently only legal in Northern Ireland if the life or mental health of the mother is at risk.
Additional reporting by Press Association