Spanish foreign minister Alfonso Dastis has said his government had been "surprised by the tone of comments coming out of Britain" about Gibraltar.
He told a Madrid conference: "It seems someone is losing their cool."
On Sunday ex-Conservative leader Lord Howard said the UK would be willing to defend Gibraltar in the same way it defended the Falklands 35 years ago.
And Gibraltar's chief minister Fabian Picardo said: "Gibraltar is not a bargaining chip in these negotiations."
"Gibraltar belongs to the Gibraltarians and we want to stay British."
The current row was sparked by draft Brexit negotiating guidelines published by the EU last Friday saying any decisions affecting Gibraltar would be run past Spain.
The guidelines said: "After the United Kingdom leaves the Union, no agreement between the EU and the United Kingdom may apply to the territory of Gibraltar without the agreement between the Kingdom of Spain and the United Kingdom."
There was a similar message from UK Foreign Secretary Boris Johnson as he arrived at EU foreign ministers' meeting in Luxembourg.
Mr Johnson said: "The sovereignty of Gibraltar is unchanged and is not going to change, and cannot conceivably change without the express support and consent of the people of Gibraltar and the United Kingdom, and that is not going to change."
Theresa May has said the UK is "committed" to the territory and its sovereignty is not on the table during Brexit talks.
The EU's guidelines followed a letter from Mrs May formally triggering Brexit talks, which did not mention Gibraltar directly.
After Mrs May spoke to Mr Picardo on Sunday, Downing Street said: "The prime minister said we will never enter into arrangements under which the people of Gibraltar would pass under the sovereignty of another state against their freely and democratically expressed wishes, nor will we ever enter into a process of sovereignty negotiations with which Gibraltar is not content."
Mr Picardo urged European Council President Donald Tusk to remove the reference to Gibraltar.
"Mr Tusk, who has been given to using the analogies of the divorce and divorce petition, is behaving like a cuckolded husband who is taking it out on the children," he said.
Defence Secretary Sir Michael Fallon told the BBC on Sunday that the UK would protect Gibraltar "all the way" because its residents had "made it very clear they do not want to live under Spanish rule".
Former Conservative leader Lord Howard, raised the spectre of military action, saying that 35 years ago, "another woman prime minister sent a taskforce halfway across the world to protect another small group of British people against another Spanish-speaking country.
"And I'm absolutely clear that our current woman prime minister will show the same resolve in relation to Gibraltar as her predecessor did."
After Argentina invaded the Falklands in 1982, Margaret Thatcher sent a task force to reclaim the islands, in the South Atlantic.
An estimated 655 Argentine and 255 British servicemen lost their lives in the fighting that followed.
Former Labour foreign secretary Jack Straw, whose 2002 referendum asking Gibraltarians if they wanted Britain to share sovereignty with Spain was rejected by 99% to 1%, dismissed the threat of military action as "frankly absurd and reeks of 19th century jingoism", adding that Britain leaving the EU would result in "all sorts of problems" popping up.
"For the Spanish, Gibraltar is an affront to their sense of national identity and their sense of sovereignty - it's a bit like having a part of Dover owned by Spain," he told BBC Radio 4's Today programme.
Mr Straw said while Britain was in the EU "we held equal cards with Spain", but once it left, the situation would be reversed, with the 27 EU nations "holding the cards".
Lib Dem leader Tim Farron said of Lord Howard's comments: "In only a few days the Conservative right are turning long-term allies into potential enemies."
But Brexit Secretary David Davis, in Spain for meetings with members of its government, said Lord Howard was expressing the "resolve" of the UK in supporting the sovereignty of Gibraltar. He made it clear any talk of Falklands style taskforces "wasn't going to happen".
Spain has long contested Britain's 300 year-rule of Gibraltar, which has a population of about 30,000.