A spokesman for the Royal Gibraltar Police confirmed that "at present" the deaths appeared to be a murder-suicide.
The spokesman said he understood that next of kin in the UK have been informed of the tragedy.
A member of staff of the company that was letting out the flat, which is a rental property, attended the residence.
They got no answer from within but noticed that the house was locked from inside and that sent a few alarm bells ringing and they decided to call the police.
At the moment what we are doing is trying to get as much background information from both the Spanish and the UK sides of the family.
These people had only been in Gibraltar for a few days, I think it was either last Thursday or Friday that they arrived.
We are trying to discover if they were here in transit to the UK or whether they were planning on staying for any length of time in Gibraltar.
A Home Office forensic pathologist is expected to travel to Gibraltar in the next couple of days to perform post-mortem examinations on the bodies.?
A family of four have been found dead in Gibraltar police have confirmed.
In a press release the Royal Gibraltar Police said a 31-year-old British man, a 37-year-old Spanish woman, a four-year-old girl and a six-week-old baby were found dead this morning.
In a press release police said: "Presently, we are not looking for anyone involved in this very unfortunate event, but the investigation will be thorough and detailed drawing in on the expertise of other professionals."
Gibraltar have parted company with manager Allen Bula four games into their maiden European Championship qualifying campaign.
Gibraltar, who became the 54th nation to join UEFA as a full member in May 2013, have lost all four of their Euro 2016 qualifiers so far, conceding 21 goals and scoring none, and are bottom of Group D.
The Gibraltar Football Association can confirm that it has parted company with national A team manager Allen Bula, with immediate effect.
After a thorough and careful investigation, into internal matters, the Gibraltar FA has come to the decision that an immediate change was appropriate. It is not a decision that has been reached lightly or hastily.
The Gibraltar FA is committed to safeguarding football on the Rock and ensuring that the national team remains the pinnacle of Gibraltarian football. Consequently the search for a new Gibraltar national team manager will begin immediately.
The Gibraltar FA would like thank Allen Bula for all the work he has done in his time as manager of the Gibraltar national team and take this opportunity to wish him all the success in the future.
Royal Gibraltar Police have advised that the "situation" at Gibraltar Airport is now over.
The area remains congested but the Winston Churchill Avenue, which links the British territory with Spain, has now reopened.
Gibraltar Airport is on alert after police said they were dealing with a "situation". At least one flight has been diverted to Malaga in Spain as a result.
A troop of 'problem' Barbary macaques have been shipped from Gibraltar to Scotland to help reduce the territory's monkey populace.Read the full story ›
A 94-year-old daredevil walked on a bi-plane as it flew around Gibraltar. Tom Lackey completed two laps strapped to the top wing.Read the full story ›
Thirty raucous residents are set to be shipped out of Gibraltar and sent to the Britain - as the UK colony admits its monkey population has got a little out of control.
The Barbary macaques are a major draw for tourists visiting the small peninsula, which is home to the only wild ape population in Europe.
But with the population now at around 300, and increasing reports of the infamous inhabitants harassing locals, the government has made the decision to deport 30 of them to a UK wildlife park.
The monkeys are thought to be to blame for damaged property, stolen bags and attacks on people.
And they are increasingly abandoning their home on the nature reserve on the Upper Rock to venture into nearby towns.
The export is set to be officially announced at the end of the week.
Previous attempts have been made to curb the number of macaques but a cull of more than 20 and the export of another 20 to a German zoo failed to help.
The Foreign and Commonwealth Office (FCO) has raised "serious concerns" after a Spanish Navy vessel wrongly claimed commercial ships off Gibraltar were in Spanish waters.
Spanish Ambassador Federico Trillo has been summoned to appear at the Foreign Office following the incident, which occurred in international waters.
The FCO says the Spanish vessel's actions constitute a breach of the UN Convention on the Law of the Sea.
Minister for Europe, the Rt Hon David Lidington MP, said: “I am extremely concerned by the actions of a Spanish Navy vessel which sought to redirect two commercial vessels heading to and from the Port of Gibraltar, wrongly claiming they were in Spanish waters.
“I condemn this provocative activity and urge the Spanish government to ensure that it is not repeated. Her Majesty’s Government will continue to take whatever action we consider necessary to uphold the interests of Gibraltar, its people, its security and economy."
Prime Minister David Cameron raised Gibraltar with Spanish Prime Minister Mariano Rajoy in the margins of an EU Council meeting.
He made it clear there must not be a repeat of last summer's disruption when the imposition of additional border checks by the Spanish led to long delays at the frontier.
People in Gibraltar feel "under siege" due to the territory being the subject of diplomatic wrangling between the UK and Spanish governments, the House of Commons' Foreign Affairs Committee has said.
"The combination of direct pressure on its border and sovereign waters as well as
diplomatic pressure at the EU, UN and NATO, have resulted in a feeling in Gibraltar of
being under siege," the cross-party committee's latest report says.
The committee accuses Spain of "aggressive policies" towards Gibraltar and says the Government "has responsibilities toward Gibraltar and cannot ignore actions by Spain that are intended to make the lives of Gibraltarians difficult".