Mr Borrell told Politico: "I am very much (more) anxious about the unity of the United Kingdom than the unity of the Kingdom of Spain".
Gibraltar, a tiny British territory located at the southern tip of the Iberian Peninsula, has always been a point of contention between the United Kingdom and Spain, with Madrid asserting a claim to the territory despite it being ceded to the Brits centuries ago after being captured from Spain.
Brexiteers have accused Spain of hypocrisy over its claim on Gibraltar as it controls two coastal territories in North Africa, Cueta and Melilla.
Spain's EU Affairs State Secretary, Luis Marco Aguiriano, said there was still time to "legally clarify" the agreement before Sunday.
Fabian Picardo, Gibraltar's Chief Minister, has accused Spain of "raising issues at the eleventh hour" is a tactic Spain has used previously while the United Kingdom were in the EU.
But May reassured British lawmakers that "we will not exclude Gibraltar from our negotiations on the future relationship". "Immigration will continue to make a positive contribution to our national life, but the difference will be this, once we have left the European Union we will be fully in control of who comes here", she said at a CBI conference in London.
The Spanish Government, he said, can't accept that Gibraltar's future be dependent on bilateral negotiations between the United Kingdom and the EU.
Jimmy Butler on adjusting to Sixers: ‘Basketball is basketball to me’
Last night's 122-119 Sixers' is being viewed as Jimmy Butler's coming out part as a vetted member of his new squad. Of course my belief in God is going to help me play the way I'm playing.
Spain's demands on Gibraltar are the latest push by European Union states to obtain more on national interests ahead of Sunday's summit of European Union leaders, but diplomats said there was little concern these would scupper prospects for a deal. The swipe came as a row over Gibraltar threatens to derail the Brexit divorce package.
EU leaders are due to ratify the draft divorce deal at a European Council meeting on Sunday but Spain said it can not support the agreement unless it is given a clear veto on Gibraltar's inclusion within the negotiations on the future relationship.
Jeremy Corbyn said it would not have the backing of Labour, meaning May lacks enough Parliamentary backers to get the deal through.
He said: 'The language of vetos and exclusions should be the language of the past.
Keeping its veto on any European Union decision on Gibraltar would give it leverage in these talks - unless it is not made explicit in this week's agreements.