The move was announced after Mr Borrell met three of Mr McCartney’s sisters in Brussels yesterday afternoon.
Catherine, Paula and Gemma McCartney spent the morning in talks with MPs explaining their search for justice in the North was running into the ground.
Catherine McCartney described how the family was confronting a “wall of silence, which is as strong today as it was on the day Robert was murdered”.
Gemma McCartney said it was a “conspiracy of silence”. Their 33-year-old brother was stabbed after a row in a bar in Belfast city centre on January 30.
Mr Borrell said the family had full parliament support in their campaign and he had been “honoured to meet the three sisters”.
“I am deeply impressed by the bravery they have shown in standing up to impunity and by the campaign for justice they are leading.
“The rule of law must prevail in all countries of the EU and in all circumstances. Justice must prevail for all victims of violence - it cannot be partial or sectarian.”
Mr Borrell added: “The doors of the European Parliament will always be open to those who need a public tribune to fight any kind of injustice and I will discuss tomorrow with the group of chairmen what we - as an institution representing all European citizens - can do to help to break the wall of silence surrounding Robert’s murder.”
Group chairmen are the leaders of the various political groupings in the parliament, who had earlier heard Catherine McCartney urge financial backing for a civil action against those the family believes were responsible for the murder.
She said police efforts and the intervention of the ombudsman had produced no results and there was “no sign of a breakthrough”.
The alternative was a civil action - but that required money the McCartneys did not have.
Mr Borrell’s said today’s talks will look into whether there are ways to use the parliament budget to help individual cases.
Labour MEP Pronsias De Rossa said it was clear the 70 people who were in the bar on the night Mr McCartney was murdered were being intimidated and feared they themselves would become victims if they came forward.
“This is a very difficult situation which is not uncommon in Northern Ireland. I know these sisters do not like to be considered brave, but it is because of their bravery that this appalling situation is being brought to the attention of the international public.”
He said offers of financial support for the sisters could be considered in time for a debate in the parliament later this month.