"This draft suggests that Spain is trying to get away with mortgaging the future relationship between the European Union and Gibraltar to its usual obsession with our homeland", Picardo said in a statement.
Britain and Spain have engaged in centuries of squabbling over the territorial status of "the rock", and the EU's stance is likely to intensify those arguments.
In last year's Brexit referendum, 96 percent voted against leaving the European Union, but they still appear set on remaining British after the vote.
The clause is likely to face a fierce backlash from London, which has always been engaged in a sovereignty row with Spain over Gibraltar and has pledged to stand by its overseas territory known as the Rock.
The draft guidelines for Brexit talks - published today (Friday 31 March) by the EU Council - include a warning that a Spanish veto will be written in to any future deals on issues like trade and free movement in relation to the British overseas territory.
But in singling out Gibraltar, the Council appears to be siding with Spain's argument that Gibraltar is a bilateral matter for the United Kingdom and Spain, despite the British Government's insistence that the Rock's future is part of the wider negotiations.
Gibraltar's border with Spain was closed by former dictator Francisco Franco in 1969 and only reopened in the 1980s.
An EU source told the BBC that Spanish diplomats lobbied to have a veto on Gibraltar's post-Brexit relationship with the EU.
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Mr Picardo said he had spoken to Foreign Secretary Boris Johnson, who he described as "implacable" in the defence of the rights of Gibraltar.
Gibraltar's chief minister Fabian Picardo has warned against Gibraltar being used as a bargaining chip during Brexit negotiations.
The successful lobbying by Spanish diplomats will inflame tensions over United Kingdom sovereignty on the Rock which has been contested by Spain since the United Kingdom took control of Gibraltar in 1713.
Spanish politicians picked up on the fact that Theresa May's Article 50 letter, which triggered the two-year Brexit process, did not mention Gibraltar, while it did mention the land border between Northern Ireland and the Republic of Ireland.
"The whole world and the whole European Union should know: this changes nothing in respect of our continued, exclusive British sovereignty", he said.
The Tory chair of the all-party parliamentary group on Gibraltar, Jack Lopresti, said: "It's predictable that given Spain's previous behaviour, they would try and use Brexit as a fig leaf for troublemaking over the status of Gibraltar". For people living in Gibraltar the outcome of the negotiations may have significant, and life-changing, consequences.