EU Commissioner Phil Hogan's position could be under threat due to a significant push to introduce more gender balance in European institutions.
Meanwhile, it has also emerged the Government has been lobbying for re-elected MEP Mairead McGuinness to be appointed President of the European Parliament.
A senior Government source said: "She has the experience, the reputation and the contacts."
There is growing demand in Brussels for more female representation in senior EU positions and member states will come under pressure to put forward women for key roles.
The move could mean the Government will be urged to replace Mr Hogan with a female commissioner when his term in office ends later this year.
"There is a big push from a lot of the candidates going for the EU commissioner job for a 50/50 gender split on the next commission," a senior Brussels source said.
The appointment of EU commissioners is the sole responsibility of each member state. However, countries that appoint women commissioners may be rewarded with more senior portfolios.
There are currently just nine women on the 28-member EU Commission.
A recent survey by the European Women's Lobby found 54pc of people believed there should be more gender balance on the commission.
Horse trading for key roles has intensified in recent weeks as EU groupings negotiate for the top jobs in Europe. The European People's Party (EPP), of which Fine Gael is a member, has been pushing for German politician Manfred Webber to be appointed as commission president.
However, he is facing tough opposition for the role from EU Brexit negotiator Michel Barnier.
Taoiseach Leo Varadkar yesterday insisted he still supported Mr Webber's candidacy despite reports the Government would eventually weigh in behind Mr Barnier.
The Taoiseach also said the presidency should be given to the EPP because the grouping has the biggest representation in the parliament after last week's elections.
"The EPP has won the most seats in the European parliament elections and our candidate for that position is Manfred Weber, who again has been to Ireland many times and has been very supportive of Irish interests," he said.
Ahead of the EU summit, the Taoiseach held a bilateral meeting with French President Emmanuel Macron during which they signed off on a plan to connect Ireland and France's energy grid.
Mr Macron is expected to put pressure on Mr Varadkar to support Mr Barnier's candidacy.
Ms McGuinness, who topped the poll in the Midlands-North-West constituency last weekend, would be certain to lose out on the European parliament presidency if Mr Webber was appointed commission president because the EPP would not be given two senior roles.
However, if Mr Barnier is appointed, Ms McGuinness's chances of taking the top job in the parliament would greatly improve.
"She is highly regarded across Europe, popular with colleagues and has a good understanding of climate change and agriculture," a Government source said.