With around 45 seats of the new 174-seat lower chamber of parliament decided, the parties were ahead of the main opposition party, the left-wing nationalist Sinn Fein.
Local media projected Fianna Fail, led by the experienced Micheal Martin, 64, would win the largest vote share and over 40 seats.
Fine Gael, whose leader Simon Harris, 38, is the outgoing prime minister (taoiseach), were in second place according to projections, with Sinn Fein in third.
To form a majority, a party or coalition requires at least 88 seats.
Both centre-right parties have repeatedly ruled out entering a coalition with Sinn Fein.The centre-left opposition Labour party is seen by Fine Gael and Fianna Fail as the most likely junior coalition party, according to local media.
Horse-trading
The Green Party was the third member of the previous coalition but its support collapsed nationwide, with all but one seat likely to be lost.
At the last general election in 2020, the pro-Irish unity Sinn Fein — the former political wing of the paramilitary Irish Republican Army — was the most popular party but could not find willing coalition partners.
That led to weeks of horse-trading, ending up with Fine Gael, which has been in power since 2011, agreeing a deal with Fianna Fail.
During the last parliamentary term, the role of prime minister rotated between the Fianna Fail and Fine Gael leaders.
The final seat numbers, which will not be confirmed until early next week, will determine whether Harris returns as taoiseach or Martin takes the role under a rotation arrangement.
The new parliament is due to sit for the first time on December 18, but following coalition talks a new government is not expected to be formed until the new year.
The three-week campaign, launched after Harris called a snap election on November 8, was dominated by rancour over housing supply and cost-of-living crises, health, public spending, and the economy