Speaking at a monthly meeting of EU foreign ministers, Borrell also accused Israeli Prime Minister Benjamin Netanyahu of using false claims of antisemitism against the International Criminal Court (ICC) for his own political ends.
The bloc is considering reviving its European Union Border Assistance Mission (EUBAM) Rafah, which has not been operational since 2007, when the Palestinian Islamist group Hamas seized full control of Gaza.
The Rafah crossing is the main entry point for aid from Egypt, and has been closed since Israeli forces took control of it from the Gazan side nearly three weeks ago.
"They gave me green light, the political green light to reactivate EUBAM, our mission in Rafah. This could play a useful role in supporting the entry of people into Gaza, in and out," Borrell told reporters after the meeting that also saw key Arab ministers join the talks.
"But this has to be done in accordance with the Palestinian Authority, the Egyptians, and obviously Israel, Israel's authorities. We are not going to do that alone. We are not going to be the outsourcers of the security in the border. We are not a security company," he said, adding that the bloc would prepare technical plans for now.
Diplomats have said the mission was unlikely to be in place before hostilities in Rafah stopped.
Israeli airstrikes killed at least 45 Palestinians in a tent camp and wounded dozens in Rafah on Sunday, prompting an outcry from global leaders, and Borrell accused Israel of pushing ahead with military action in southern Gaza despite the ICJ ruling.
Speaking after the ministers also held talks with counterparts from Saudi Arabia, Qatar, Egypt, Jordan and the United Arab Emirates, Borrell said the two parties would look to see if they could combine their efforts to set up a joint conference on how to implement a two-state solution.
SWIPE AT NETANYAHU
Two international courts, both based in the Hague, have reached decisions opposed by Israel in recent weeks.
The prosecutor of the ICC said he was seeking the arrest of Netanyahu and Israel's defence minister along with Hamas leaders over alleged war crimes, while the International Court of Justice (ICJ), which hears disputes between states, ordered Israel to halt its offensive on Rafah.
Israel has denounced the ICC as a rogue court and called on allies to reject it. It also says its assault on Rafah does not need to be halted in line with the ICJ ruling because it does not pose an unlawful threat to civilians.
Borrell also said the bloc had agreed for the first time to hold a meeting of the EU-Israel Council - the body that coordinates trade between the two - to discuss the situation in Gaza, respect of human rights and the impact of the ICJ ruling on the association agreement.
Borrell took a swipe at Netanyahu over his labelling of ICC Chief Prosecutor Karim Khan decision's to seek arrest warrants against him and his defence minister as evidence of "new antisemitism".
Borrell described the comment as intimidation, saying acccusations of antisemitism were made every time that anyone "does something that Netanyahu doesn't like".
